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^1 I 









HYDRODYNAMICS 



CAMBRIDGE UNIVERSITY PRESS 

C F. CLAY, Manager 

ftonlron: FETTER LANE, E.C. 

eHinllttrglft : xoo PRINCES STREET 







lUiD «orli: G. P. PUTNAM'S SONS 

ISomftat, Calcutta anti iKaDTas: MACMILLAN AND CO. ,1 Ltd. 

Corotttu: J. M. DENT AND SONS, Ltd. 

ffoitso: THE MARUZENKABUSHIKIKAISHA 



Ail rights teservifd 









>f*^ 



^ 



HYDRODYNAMICS 



BY 



HORACE LAMB, M.A., LL.D., Sc.D., F.R.S. 

PROFESSOR OF MATHEMATICS IN THE VICTORIA UNIVERSITY OF MANCHESTER; 
FORMERLY FELLOW OF TRINITY COLLEGE, CAMBRIDGE 



FOURTH EDITION 



« > 

• • • 

"4 • • • 
• • . 



Cambridge : 

at the University Press 

1916 



First EdiUan 1879 
Second Edition 1896 

Third EdiUon 1906 
Fourth Edition 1916 






* 






\ 






I 



TO 

HENRY MARTYN TAYLOR 

THIS BOOK IS INSCRIBED BY HIS FRIEND 

THE AUTHOR 



PREFACE 

rilHIS book may be regarded as a fourth edition of a Treatise on the 
"*- Mathematical Theory of the Motion of Fluids^ which was published in 
1879. The second edition, largely remodelled and extended, appeared 
under the present title in 1895, and was followed by a third in 1906. In 
this issue, as in the preceding one, no change has been made in the general 
plan and arrangement, but the work has been carefully revised, occasional 
passages have been rewritten, and various interpolations and additions have 
been made, more especially in the latter part of the book, which deals mainly 
with physical applications. A few investigations of secondary interest have 
been condensed or omitted. 

A word or two may be said with regard to certain departures from general 
usage which are to be found in the book. The use of the reversed sign for 
the velocity-potential {<f>), which was adopted in the 1895 edition and is here 
continued, was not altogether an innovation, and has strong arguments of 
a physical kind to recommend it. It appears so much more natural to regard 
the state of motion of a dynamical system, in any given configuration, as 
specified by the impulses which would start it, rather than by those which 
would stop it, that the altered definition of the function would seem to require 
no further justification. It has also the advantage that the analogies with 
other branches of Mathematical Physics are rendered more complete. 

In the present edition I have been led to make a further slight change from 
prevalent usage, by choosing for special designation the vector (|, rq, t) whose 
components in terms of the velocity (w, v, w) are 

dw dv du dw dv du 
dy dz' dz dx* dx dy ' 

rather than that whose components have half these values. This procedure 
avoids the insertion of an unnecessary factor 2 or J in a number of formulae, 
in particular in the theorem of Stokes (Art. 32) which is the fundamental 



BEQUEST OF 
ALEXANDER ZIWIT« 



Preface vii 

relation in the present connection. The vector (^, r), ^) as now defined may 
conveniently be called the *vorticity,' the term * rotation' being used, if 
required, in its established sense. It may be added that the altered notation 
is in conformity with the physical analogies already referred to. It is more- 
over already current in some writings on the present subject. 

Pains have been taken to make due acknowledgment of authorities in 
the footnotes ; but it will be understood that the original methods have not 
always been followed in the text. 

I would add that the work has less pretensions than ever to be regarded 
as a complete account of the science with which it deals. The subject has 
of late attracted increased attention in various countries, and it has become 
correspondingly difficult to do justice to the growing literature. Some 
memoirs deal chiefly with questions of mathematical method and so fall 
outside the scope of this book; others though physically important hardly 
admit of a condensed analysis; others, again, owing to the multiplicity of 
publications, may imfortunately have been overlooked. And there is, I am 
afraid, the inevitable personal equation of the author, which leads him to 
take a greater interest in some branches of the subject than in others. 

I am much indebted to the staff of the University Press for their careful 
supervision of the printing, and for kindly calling attention to various over- 
sights. 

It is again a satisfaction to me to inscribe on the fly-leaf the name of 
Mr H. M. Taylor, whose kindly encouragement first led me to write on the 
subject, and whose help in revision I had gratefully to acknowledge on former 
occasions. 

HORACE LAMB. 



January, 1010. 



CONTENTS 



ABT. 
1. 
2. 

3>9. 

10. 
11. 
12. 
13, 14. 

16. 
16. 



17. 

18, 19. 
20. 

21-23. 

24. 
26. 
26-29. 



30. 

31, 32. 
33. 



CHAPTER I 

THE EQUATIONS OF MOTION 

PAGE 

Fundamental property of a fluid 1 

The two plans of investigation 2 

*Eulerian' form of the equations of motion. Dynamical equations. 

Equation of continuity. Physical equations. Surface conditions 2 

Equation of energy 8 

Impulsive generation of motion 11 

Equations referred to moving axes 12 

'Lagrangian' form of the dynamical equations and of the equation of 

continuity 13 

Weber's transformation 14 

Extension of the Lagrangian notation 16 

CHAPTER II 

INTEGRATION OP THE EQUATIONS IN SPECIAL CASES 

Velocity-potential. Lagrange's theorem 16 

Physical and kinematical relations oi<f> 17 

Integration of the equations when a velocity-potential exists; pressure- 
equation 18 

Steady motion. Deduction of the pressure-equation £rom the principle 

of energy. Limit to the velocity 19 

Efflux of liquids; vena contracta 22 

Efflux of gases 24 

Examples of rotating fluid. Uniform rotation. Bankine's * combined 

vortex.' Electromagnetic rotation 26 

CHAPTER III 

IRROTATIONAL MOTION 

Analysis of the differential motion of a fluid element into strain and 

rotation 29 

*Flow' and * circulation.' Stokes' theorem 31 

Constancy of circulation in a moving circuit 34 



Contents 



IX 



ABT. PAGE 

34, 35. Irrotational motion in simply-connected spaces; single- valued velocity- 
potential 35 

36-30. Incompressible fluids; tubes of flow. <f> cannot be a maximum or 

minimum. The velocity cannot be a maximum. Mean value of <f> 

over a spherical surface 36 

40, 41. Conditions of determinateness of ^ 39 

42-46. Green's theorem ; dynamical interpretation; formula for kinetic energy. 

Kelvin's theorem of minimum energy 42 

47, 48. Multiply-connected regions; 'circuits' and 'barriers' .... 47 
49-51. Irrotational motion in multiply-connected spaces ; many-valued velocity 

potential; cyclic constants 48 

52. Case of incompressible fluids. Conditions of determinateness of ^ . . 51 

53-55. Kelvin's extension of Green's theorem; dynamical interpretation; 

energy of an irrotationally moving liquid in a cyclic space . 52 

56-58. 'Sources' and 'sinks.' Double sources. Irrotational motion of a liquid 

in terms of surface-distributions of sources 55 



CHAPTEE IV 



MOTION OP A LIQUID IN TWO DIMENSIONS 

59. Lagrange's stream-function 60 

60-62. Relations between stream- and velocity-functions. Connection with the 

theory of complex variables 62 

63, 64. Simple types .of motion, acyclic and cyclic. Potential of a row of simple 

or double sources 66 

65, 66. Inverse relations. Examples; confocal curves; flow tcom an open 

channel 69 

67. General formulae; Fourier method 72 

68. Motion of a circular cylinder without circulation; stream-lines . . 73 

69. Motion of a cylinder with circulation. Trochoidal path under constant 

force 75 

70. Note on more general problems . . . . * . . . . 78 

71. Inverse methods. Motion due to the translation of a cylinder; case of 

an elliptic cylinder. Flow past an oblique lamina; couple-resultant 

of fluid pressures 78 

72. Motion due to a rotating rigid boundary. Rotating prismatic vessel whose 

section is an ellipse, equilateral triangle, or circular sector. Rotating 
eUiptic cylinder in infinite liquid. Formula for the most general 
motion of an elliptic cylinder with circulation .... 82 

73. Steady motions with a free surface. Schwarz' method of conformal 

transformation 86 

74. Two-dimensional form of Borda's mouthpiece 88 

75. Fluid issuing from a rectilinear aperture. Coefficient of contraction . 90 

76. 77. Impact of a stream on a lamina, direct and obUque. Calculation of 

resistance 92 

78. .Bobyleff's problem 96 

79. Discontinuous motions 98 

80. Flow in a curved stratum 101 



X Contents 



CHAPTER V 

IRROTATIONAL MOTION OF A LIQUID: PROBLEMS IN 

THREE DIMENSIONS 

ART. PAGE 

81, 82. Spherical harmonics. Maxwell's theory of poles 103 

83. Laplace*8 equation in polar co-ordinates 105 

84, 85. Zonal harmonics. Hypergeometric series 106 

86. Tesseral and sectorial harmonics 109 

87, 88. Conjugate property of surface-harmonics. Expansions . . . Ill 

89. Symbolical solutions of Laplace's equation. Definite- integral forms . 112 

90, 91. Hydrodynamical applications. Impulsive pressure over a spherical 

surface. Prescribed normal velocity. Energy of motion generated 114 
92, 93. Motion of a sphere in an infinite liquid. Inertia coefficient. Sphere in 

a liquid with concentric spherical boundary 115 

94-96. Stokes* stream-function. Expression in spherical harmonics. Stream- 
lines of a sphere. Image of a double source in a sphere . 118 

97. Bankine's inverse method 122 

98, 99. Motion of twq spheres in a liquid; kinematical formulae . . . 123 
100, 101. Cylindrical harmonics. Solutions of Laplace's equation in terms of 

BessePs functions. Expansion of an arbitrary function . . 127 
102. Hydrodynamical examples. Flow through a circular aperture. Inertia 

coefficient of a circular disk 130 

103-106. Ellipsoidal harmonics for an ovary ellipsoid. Solutions of Laplace's 

equation. Applications to the motion of an ovary eUipsoid in liquid 133 
107-109. Harmonics for a planetary ellipsoid. Flow through a circular aperture. 

Stream-lines of a circular disk. Translation and rotation of a plane- 
tary ellipsoid 137 

110. Motion of a fluid in an ellipsoidal envelope 141 

111. General expression for V'(^ in orthogonal co-ordinates .... 142 

112. Confocal quadrics; ellipsoidal co-ordinates 143 

113. Flow through an elliptic aperture 145 

114,115. Translation and rotation of an ellipsoid in liquid ; inertia coefficients . 147 

116. References to other problems 150 



CHAPTEE VI 

ON THE MOTION OF SOLIDS THROUGH A LIQUID: 

DYNAMICAL THEORY 

117, 118. Kinematical formulae for the case of a single body . . . .151 

119. Theory of the 'impulse' 153 

120. Dynamical equations relative to axes fixed in the solid . .154 

121. Kinetic energy; coefficients of inertia 155 

122. 123. Components of impulse. Reciprocal formulae 156 

124. The three permanent translations; stability 158 

125. The possible types of steady motion. Motion due to an impulsive couple 160 

126. Hydrokinetic symmetry .... 162 



Contents xi 

ART. PAGE 

127-129. Motion of a solid of revolution. Stability of motion parallel to an axis 

of symmetry. Influence of rotation. Other types of steady motion 165 

130. Motion of a 'heUcoid' 170 

131. Inertia-coefficients of a liquid contained in a moving rigid envelope . 171 
132-134. Case of a perforated solid with cyclic motion through the apertures. 

Steady motion of a ring; condition of stability . . . .171 

135. Lagrange's equations of motion in generalized co-ordinates. Hamiltonian 

principle 175 

136. Adaptation to hydrodynamics 178 

137. 138. Motion of a sphere near a rigid boundary. Motion of two spheres in the 

line of centres 181 

130. Modification of Lagrange's equations in the case of cyclic motion; 

ignoration of co-ordinates 183 

140, 141. Equations of motion of a gyrostatic system 186 

142. Kineto-statics 189 

143, 144. Motion of a sphere in a cyclic region. Circulation round thin cores or 

through tubes; comparison with electromal^netic phenomena 191 



CHAPTER VII 

VORTEX MOTION 

146. * Vortex-lines' and * vortex-filaments'; kinematical properties . . 194 

146. Persistence of vortices; Kelvin's proof. Equations of Cauchy, Stokes, 

and Helmholtz. Motion in a fixed ellipsoidal vessel, with uniform 

vorticity 196 

147. Conditions of determinateness in vortex motion 200 

148. 149. Velocity expressed in terms of expansion and vorticity; electromagnetic 

analogy. Case of an isolated vortex 201 

150. Velocity-potential due to a vortex 204 

151. Vortex-sheets 206 

152. Impulse of a vortex-system 208 

153. Formulae for the kinetic energy 209 

154. 155. Rectilinear vortices. Stream-lines of a vortex-pair. Other examples . 213 

156. Investigation of the stability of a row of vortices ; and of a double row . 218 

157. Creneral formulae relating to a rectilinear vortex-system. Kirchhoff's 

theory 223 

158. Stability of a cylindrical vortex 224 

159. Kirchhoff's eUiptic vortex 226 

160. Vortices in a curved stratum of fluid 227 

161-163. Circular vortices. Potential- and stream-function of an isolated vortex- 
ring; stream-lines. Impulse and energy; velocity of translation 

of a vortex-ring 227 

164. Mutual influence of vortex-rings. Image of a vortex-ring in a sphere 233 

165. General conditions for steady motion of a fluid. Cylindrical and 

spherical vortices 235 

166. References 238 

167. Clebsch's transformation of the hydrodynamical equations . . . 239 



xii Contents 



CHAPTER VIII 

TIDAL WAVES 
ABT. PAOB 

168. General theory of small oscillations; normal modes ; forced oscillations 241 

169-174. Free waves in uniform canal ; wave-velocity ; effect of initial conditions ; 

physical meaning of the various approximations; energy of a wave- 
system 246 

176. Artifice of steady motion 253 

176. Superposition of wave-systems; reflection 253 

177-179. Effect of disturbing forces ; free and forced oscillations in a finite canal 254 
180-184. Canal theory of the tides ; disturbing potential. Tides in an equatorial 

canal, and in a canal parallel to the equator; semi-diurnal and 
diurnal tides. Canal coincident with a meridian ; change of mean 
level; fortnightly tide. Equatorial canal of finite length; lag of 

the tide 258 

185, 186. Waves in a canal of variable section. Examples of free and forced 

oscillations; exaggeration of tides in shallow seas and estuaries . 265 
187, 188. Waves of finite amplitude ; change of type in a progressive wave. Tides 

of the second order 269 

189, 190. Wave-motion in two horizontal dimensions; general equations. Oscil- 
lations of a rectangular sheet of water 274 

191,192. Oscillations of a circular sheet; Bessel's functions ; contour-lines . . 276 

193. Case of variable depth. Circular basin 283 

194, 195. Propagation of disturbances outwards from a centre; Bessel's function 

of the second kind. Waves due to a periodic local pressure . 284 
196, 197. General formula for diverging waves. Example of a transient local 

disturbance 290 

198-201. Oscillations of a spherical sheet of water ; free and forced waves. Effect 

of the mutual attraction of the liquid. Reference to the case of a sea 

bounded by meridians and parallels 294 

202, 203. General equations of motion of a d3mamical system relative to rotating 

axes 300 

204. Application to the small oscillations of a rotating system . . . 302 

205, 206. Free oscillations ; stability, 'ordinary* and 'secular.' Forced oscillations 303 
207, 208. Application to hydrod3mamics ; tidal oscillations of a rotating plane 

sheet of water; waves in a straight canal 308 

209-211. Circular sheet of uniform depth; free and forced oscillations . .311 

212. Circular basin of variable depth 316 

213, 214. Tidal oscillations on a rotating globe. Equations of Laplace's kinetic 

theory 318 

215-217. Case of symmetry about the axis. Tides of long period . . .321 
218-221. Diurnal and semi-diurnal tides. Discussion of Laplace's solution . . 328 
222,223. Hough's investigations; extracts and results 335 

224. Modifications of the kinetic theory due to the actual configuration of the 

ocean; questions of phase 341 

225, 226. Stability of the ocean. Remarks on the general theory of kinetic 

stability 343 

Appendix: On Tide-generating forces 346 



Contents xiii 



CHAPTER IX 

SURFACE WAVES 

ABT. PAOB 

227. Statement of the two-dimenflional problem; surface-conditions . . 351 

228. Standing waves; lines of motion 352 

229, 230. Progressive waves ; orbits of particles. Wave- velocity ; numerical 

tables. Energy of a simple-harmonic wave-train . . . . 354 

231, 232. Artifice of steady motion. Oscillations of superposed liquids . 359 

233, 234. Waves on the boundary between two currents ; instability . . 363 

235. Waves in a heterogeneous liquid 367 

236, 237. Theory of 'group- velocity.' Propagation of energy .... 369 
238-240. The Cauchy-Poisson wave-problem; waves due to an initial local 

elevation, or to a local impulse 373 

241. Kelvin's approximate formula for the effect of a local disturbance 

in a medium; graphical constructions 384 

242-246. Surface-disturbance of a stream. Case of finite depth. Effect of 

inequalities in the bed 388 

247. Waves due to a submerged cylinder 401 

248. General theory of waves due to a travelling disturbance; approximate 

formidae 403 

249. Wave-resistance. Example of the submerged cylinder . . . 407 

250. Surface-waves of finite height. Stokes' waves of permanent type. Limiting 

form 409 

251. Gerstner's rotational waves 412 

252. 253. Solitary waves. Oscillatory waves of Korteweg and De Vries . . 415 

254. Hebnholtz' dynamical condition for waves of permanent type . . 420 

255. Wave-propagation in two horizontal dimensions. Effect of a localized 

initial disturbance 422 

256. Effect of a pressure-disturbance advancing over the surface of water; 

wave-pattern; ship- waves 426 

257. Standing waves in limited masses of water. Case of uniform depth. 

Oscillations in a rectangular tank with a cylindrical obstacle . 432 

258. 259. Transverse oscillations in channels of triangular and semicircular section 435 
260,261. Longitudinal oscillations; cases of triangular section. Edge- waves . 438 
262-264. Oscillations of a liquid globe; lines of motion. Ocean of uniform depth 

on a spherical nucleus 443 

265. Capillarity; surface-condition 448 

266. Capillary waves; group- velocity 449 

267. Waves under both gravity and capillarity; minimum wave- velocity . 452 

268. Waves on the common boundary of two currents .... 455 

269. Waves due to a local disturbance. Effect of a travelling disturbance; 

waves and ripples 456 

270,271. Surface-disturbance of a stream; formal investigation . . 457 

272. Effect of a pressure-point advancing over the surface of water; fish-line 

problem; wave-patterns 462 

273, 274. Vibrations of a cylindrical column of liquid; instability of a jet . 465 
275. Oscillations of a liquid globule, and of a bubble 468 



xiv Contents 



CHAPTER X 

WAVES OP EXPANSION 
ABT. PAOB 

276-280. Plane waves; velocity of sound; energy of a wave-system . . 470 

281-284. Plane waves of finite amplitude; methods of Biemann and Eamshaw. 

Condition for permanence of type ; Bankine's investigation. Question 
as to the possibility of a wave of discontinuity .... 475 

285, 286. Spherical waves. Solution in terms of initial conditions . . . 483 
287, 288. General equation of sound-waves. Poisson's integral Equation of 

energy. Determinateness of solutions 486 

289. Simple-harmonic vibrations. Sources of sound, simple and double; 

emission of energy 489 

290. Helmholtz' adaptation of Green' s theorem. Velocity-potential in terms of 

surface-distributions of sources. Kirchhoff*s formula . . .491 

291. Periodic disturbing forces 495 

292. Applications of spherical harmonics; general formulae . . . 497 

293. Vibrations of air in a spherical envelope. Vibrations of a spherical 

stratum 500 

294. Propagation of waves outwards from a spherical surface ; effect of lateral 

motion 502 

295. Theory of the ball-pendulum; correction for inertia; coefficient of 

dissipation (due to air-waves) 504 

296-298. Scattering of sound-waves by a sphere. Impact of waves on a moveable 

sphere; case of s3mchronism 505 

299, 300. Approximate treatment of diffraction problems when the wave-length is 

relatively large. Diffraction by a flat disk, by an aperture in a thin 

screen, and by a small obstacle of any form 510 

301. Solution of the equation ^ =c^v^<f> in spherical harmonics. Conditions 

at the front of a diverging wave 517 

302. Sound-waves in two dimensions ; effect of a transient source ; comparison 

with the one- and three-dimensional cases 519 

303. Simple-harmonic vibrations ; solutions in Bessel's functions. Oscillating 

cylinder 523 

304. Scattering of waves by a cylindrical obstacle 525 

305. Approximate theory of diffraction of long waves in two dimensions. 

Diffraction by a flat blade, and by an aperture in a thin screen . 527 

306. 307. Reflection and transmission of sound-waves by a grating . . . 529 

308. Diffraction by a semi-infinite screen 535 

309, 310. Waves propagated vertically in the atmosphere; 'isothermal' and 

*convective' hypotheses 538 

311. General equations of atmospheric waves 544 

312, 313. Two-dimensional case; gravitational oscillations 546 

314-316. Large-scale oscillations of an atmosphere surrounding a globe, without. 

and with rotation. Atmospheric tides 551 



Contents xv 



CHAPTER XI 

VISCOSITY 
ABT. PAGE 

317y 318. Theory of dissipative forces. One degree of freedom; free and forced 

oscillations. Effect of friction on phase 556 

319. Application to tides in equatorial canal; tidal lag, and tidal friction . 559 

320. Equations of dissipative systems in general; frictional and gyrostatic 

terms. Dissipation function 563 

321. Oscillations of a dissipative system about a configuration of absolute 

equilibrium 564 

322. Effect of gyrostatic terms. Example of two degrees of freedom; dis- 

turbing force of long period 565 

323-325. Viscosity of fluids ; specification of stress; formulae of transformation . 567 
326, 327. The stresses assumed to be linear functions of rates of strain. Coefficient 

of viscosity. Boundary conditions; question of slipping . . 569 

328. Dynamical equations. The modified Helmholtz equations; interpretation 572 

329. Dissipation of energy in a liquid by viscosity 574 

330. Problems of steady motion: flow of a liquid between parallel planes; 

Hele Shaw's experiments 576 

331. 332. flow through a pipe of circular section; Poiseuille's laws; question of 

slipping. Other forms of section 577 

333, 334. Steady rotation of a cylinder, and of a sphere. Practical limitation 

to the solutions 580 

335, 336. General solution of the problem of slow steady motion in spherical 

harmonics. Formulae for the stresses 583 

337. Rectilinear motion of a sphere; resistance; terminal ^velocity ; stream- 

lines. Motion of a liquid sphere. Motion of a solid sphere, with 
slipping 586 

338. Method of Stokes; solutions in terms of the stream-function . .591 

339. Steady motion of an ellipsoid 503 

340. Questions as to the validity of preceding solutions. Oseen's criticism. 

Modified investigation 595 

341. 342. Steady motion of a liquid in a given field of force. Analogy with the 

theory of flexure of elastic plates 601 

343. Steady motion of a cylinder, treated by Oseen's method. Resistance . 604 

344. Dissipation of energy in steady motion; theorems of Helmholtz and 

Korteweg. Rayleigh's extension 606 

345-347. Problems of periodic motion. Laminar motion; diffusion of vorticity. 

Oscillating plane. Periodic tidal force ; feeble influence of viscosity 

on rapid motions 609 

348, 349. Effect of viscosity on water-waves 614 

350, 351. Effect of surface-forces ; generation and maintenance of waves by wind. 

Scott Russell's observations. Calming effect of oil . . . 619 

352, 353. Periodic motion with a spherical boundary ; general solution in spherical 

harmonics 622 

354. Applications; decay of motion in a spherical vessel; torsional oscilla- 
tions of a hollow sphere containing liquid 628 



^^^^tmmm^m^^am^^^rmmma^mimim^mmmmmmu^tmmmmmf'a^''Bmmmr^!^ u jl 



xvi Contents 

ABT. PAGE 

356. Osoillations of a liquid globe 630 

356. Effect of viscoeity on the rotational oscillations of a sphere, and on the 

pendulum 632 

357. Notes on two-dimensional problems 635 

358. Viscosity in gases; dissipation function 636 

359. Damping of plane waves of sound by viscosity 637 

360. Combined influence of viscosity and thermal conduction . . 639 

36L Effect of viscosity on diverging waves 641 

362, 363. Effect on the scattering of sound-waves by a spherical obstacle, fixed or 

free 645 

364. Damping of sound-waves in a spherical vessel 649 

365, 366. Turbulent motion. Be3mold's experiments ; critical velocities of water 

in a pipe; law of resistance. Inferences from theory of dimensions 651 

367, 368. References to theoretical investigations of Bayleigh and Kelvin . . 655 

369. Statistical method of Be3molds 660 

370-372. Resistance of fluids. References to various theories. 'Peripteroid' 

motion. Dimensional formulae 664 



CHAPTER XII 

ROTATING MASSES OF LIQUID 

373. Formulae relating to attraction of ellipsoids. Potential energy of an 

ellipsoidal mass 669 

374. Maclaurin's ellipsoids. Relations between eccentricity, angular velocity 

and angular momentum; numerical tables 671 

375. Jacobi's ellipsoids. Linear series of ellipsoidal forms of equilibrium. 

Numerical results 673 

376. Other special forms of relative equilibrium. Rotating annulus . . 677 

377. General problem of relative equilibrium; Poincar6's investigation. 

Linear series of equilibrium forms; limiting forms and forms of 

bifurcation. Exchange of stabilities 680 

378-380. Application to a rotating system. Secular stability of Maclaurin's and 

Jacobi's ellipsoids. The pear-shaped figure of equilibrium . . 683 

381. Small oscillations of a rotating ellipsoidal mass; Poincar6*s method. 

References 687 

382. Dirichlet's investigations; references. Finite gravitational oscillations 

of a liquid ellipsoid without rotation. Oscillations of a rotating 
ellipsoid of revolution 689 

383. Dedekind^s ellipsoid. The irrotational ellipsoid. Rotating elliptic 

cylinder 692 

384. Free and forced oscillations of a rotating ellipsoidal shell containing 

liquid. Precession 694 

385. Precession of a liquid ellipsoid 698 

List of Authobs cited 702 

Index 705 



H YDEOD YN AMIC S 



CHAPTER I 

THE EQUATIONS OF MOTION 

1. The following investigations proceed on the assumption that the 
matter with which we deal may be treated as practically continuous and 
homogeneous in structure ; i.e. we assume that the properties of the smallest 
portions into which we can conceive it to be divided are the same as those of 
the substance in bulk. 

The fundamental property of a fluid is that it cannot be in equilibrium in 
a state of stress such that the mutual action between two adjacent parts is 
oblique to the common surface. This property is the basis of Hydrostatics, 
and is verified by the complete agreement of the deductions of that science 
with experiment. Very slight observation is enough, however, to convince 
us that obUque stresses may exist in fluids in motion. Let us suppose for 
instance that a vessel in the form of a circular cyUnder, containing water 
(or other liquid), is made to rotate about its aids, which is vertical. If the 
angular velocity of the vessel be constant, the fluid is soon found to be rotat- 
ing with the vessel as one soUd body. If the vessel be now brought to rest, the 
motion of the fluid continues for some time, but gradually subsides, and at 
length ceases altogether ; and it is found that during this process the portions 
of fluid which are further from the aids lag behind those which are nearer, 
and have their motion more rapidly checked. These phenomena point to the 
existence of mutual actions between contiguous elements which are partly 
tangential to the common surface. For if the mutual action were everywhere 
wholly normal, it is obvious that the moment of momentum, about the axis 
of the vessel, of any portion of fluid bounded by a surface of revolution about 
this axis, would be constant. We infer, moreover, that these tangential 
stresses are not called into play so long as the fluid moves as a solid body, 
but only whilst a change of shape of some portion of the mass is going on, 
and that their tendency is to oppose this change of shape. 

L. H. 1 



The Equations of Motion 



[chap. I 




2. It is usual, however, in the first instance to neglect the tangential 
stresses altogether. Their effect is in many practical cases small, and inde- 
pendently of this, it is convenient to divide the not inconsiderable difficulties 
of our subject by investigating first the effects of purely normal stress. The 
further consideration of the laws of tangential stress is accordingly deferred 
till Chapter xi. 

If the stress exerted across any small plane area situate at a point P of 
the fluid be wholly normal, its intensity (per 
unit area) is the same for all aspects of the 
plane. The foUowing proof of this theorem 
is given here for purposes of reference. 
Through P draw three straight lines PA, 
PB, PC mutually at right angles, and let 
a plane whose direction-cosines relatively to 
these lines are {, m, n, passing infinitely 
close to P, meet them in A, By C. Let 
p, pi, p^f p^ denote the intensities of the 

stresses* across the faces ABC, PBC, PC A, PAB, respectively, of the 
tetrahedron PABC, If A be the area of the first-mentioned face, the areas 
of the others are, in order, ?A, mA, nA. Hence if we form the equation of 
motion of the tetrahedron parallel to PA we have p^.lA^ pi . A, where we 
have omitted the terms which express the rate of change of momentum, and 
the component of the extraneous forces, because they are ultimately propor- 
tional to the mass of the tetrahedron, and therefore of the third order of 
small linear quantities, whilst the terms retained are of the second. We 
have then, ultimately, p== Pi, and similarly p = p^ = p^^ which proves the 
theorem. 

3. The equations of motion of a fluid have been obtained in two different 
forms, corresponding to the two ways in which the problem of determining 
the motion of a fluid mass, acted on by given forces and subject to given 
conditions, may be viewed. We may either regard as the object of our 
investigations a knowledge of the velocity, the pressure, and the density, 
at all points of space occupied by the fluid, for all instants ; or we may seek 
to determine the history of every particle. The equations ^obtained on these 
two plans are conveniently designated, as by German mathematicians, the 
'Eulerian' and the 'Lagrangian' forms of the hydrokinetic equations, 
although both forms are in reality due to Eulerf. 

* ReokoDed positiTe when preesures, negatiTe when tenaioDB. Most fluids are, however, 
incapable under otdinaiy conditions of supporting more than an exceedingly slight degree of 
tension, so that p is nearly always positive. 

t "Principes g6n6ranx du mouvement des floides," Hist, de VAcad, de Berlin, 1755. 

"De principxis motus fluidorum," Nwi Comm, Aead, Pelrap. t. xiv. p. 1 (1759). 

Lagrange gave three investigations of the equations of motion; first, incidentally, in 



2-5] Btderian Equations 8 

The Eulerian Equations. 

4. Let u, V, w be the components, parallel to the co-ordinate axes, of the 
velocity at the point (x, y^ z) at the time t. These quantities are then 
functions of the independent variables Xy y, z, t. For any particular value of 
t they e^^press the motion at that instant at all points of space occupied by 
the fluid; whilst for particular values of x, y, z they give the history of 
what goes on at a particular place.' 

We shall suppose, for the most part, not only that ti, v, w are finite and 
continuous functions of x, y, z, but that their space-derivatives of the first 
order {du/dx, dv/dx, dw/dx, &c.) are everywhere finite*; we shall understand 
by the term 'continuous motion,' a motion subject to these restrictions. 
Cases of exception, if they present themselves, will require separate examina- 
tion. In continuous motion, as thus defined, the relative velocity of any two 
neighbouring particles P, P' wiU always be infinitely smaU, so that the line 
PP' will always remain of the same order of magnitude. It follows that if 
we imagine a small closed surface to be drawn, surrounding P, and suppose 
it to move with the fluid, it will always enclose the same matter. And any 
surface whatever, which moves with the fluid, completely and permanently 
separates the matter on the two sides of it. 

5. The values of u, v, w for successive values of t give as it were a series 
of pictures of consecutive stages of the motion, in which however there is no 
immediate means of tracing the identity of any one particle. 

To calculate the rate at which any function F (x, y, z, t) varies for a 
moving particle, we remark that at the time t + St the particle which was 
originally in the position {x, y, z) is in the position {x + 1^8^ y + vht^ z + wU), 
so that the corresponding value of P is 

7^W f^JP rUP rUP 

P (a: -t- w8<, y 4- v8t, z + wU, <4.&)«P-fw8i — -hi;&— +w;&|i4.8<^. 

If, after Stokes, we introduce the symbol D/Dt to denote a differentiation 
foUowing the motion of the fluid, the new value of P is also expressed by 
P + DF/Dt . 8t, whence 

DF ap. ap. ap. ap 

:D^ = -a7+^ax+"a^ + ^a^ (^) 

connection with the princijde of Least Action, in the MisceUanea Taurinenata, t. ii. (1760) 
[Oeuvres, Paris, 1807-92, t. i.]; secondly in his "M^moife snr la Theorie du Honvement dee 
Fioides," Nouv. ttUm. de VAcad. de Berlin^ 1781 [Otuvres, t. iv.]; and thirdly in the Micaniq%e 
Anahftique. In this last exposition he starts with the second form of the equations (Art. 14, 
below), but translates them at once into the * Eulerian ' notation. 

* It is important to bear in mind, with a view to some later developments mider the head 
of Vortex Motion, that these derivatiTes need not be assumed to be continuous. 

1—2 



The Equations of Motion 



[chap. I 



6. To form the dynamical equations, let p be the pressure, p the density, 
Xy y, Z the components of the extraneous forces per unit mass, at the point 
(x, y, z) at the time L Let us take an element having its centre at {Xy y, z), 
and its edges 8x, By, Sz parallel to the rectangular co-ordinate axes. The rate 
at which the rc-component of the momentum of this element is increasing is 
p8xBy8zDu/Dt; and this must be equal to the a^-component of the forces 
acting on the element. Of these the extraneous forces give pSxSySzX. The 
pressure on the yz-iace which is nearest the origin will be ultimately 

that on the opposite face 

{P + h^Pl^^ • Src) Syhz. 

The difference of these gives a resultant — dp/dx , SxSySz in the direction of 
x-positive. The pressures on the remaining faces are perpendicular to x. 
We have then 

phx8yiz-jr- = pSxSySzX — ^8xSy8z. 

Substituting the value of Du/Dt from (1), and writing down the 
symmetrical equations, we have 

du 



du du du du ^ 1 dp 



pdx' 
dv dv dv dv ^, 1 dp 



dx 

dv 
dx 



dy 

dv 

dy 



dz 

dv 
Wz 



pdy' 



y 



dw , dw , dw , dw „ ^dp 



(2) 



7. To these dynamical equations we must join, in the first place, a 
certain kinematical relation between w, v, w, p, obtained as follows. 

If V be the volume of a moving element, we have, on account of the 
constancy of mass,*/ 



D.pv 
Dt 



= 0, 



or 



1^+1^ = 



(1) 



To calculate the value of 1/v . D^/Dt, let the element in question be that 
which at time t fills the rectangular space SxSySz having one corner P at 
(x, y, z), and the edges PL, PM, PN (say) parallel to the co-ordinate axes. 
At time t -\- Bt the same element will form an oblique parallelepiped, and since 



* It is easily seen, by Taylor's theorem, that the mean pressure over any face of the element 
dxSy&t may be taken to be eqnal to the pressure at the centre of that face. ^ ... / 



V0<« 



6-7J Uqtiation of Continuity 5 

the velocities of the particle L relative to the particle P are dujdx . Sx, 
dvjdx . Sx, dwjdx . 8x, the projections of the edge PL become, after the time 8f, 

respectively. To the first order in &, the length of this edge is now 

(l+|«)8.. 

and similarly for the remaining edges. Since the angles of the parallelepiped 
differ infinitely little from right angles, the volume is still given, to the first 
order in 8i, by the product of the three edges, i.e. we have 

\D^ du dv dw ,n\ 

°' ^Dt'di^Ty^Tz (^^ 

Hence (1) becomes 

Dp (du dv dw\ .ox 

This is called the * equation of continuity.' 

The expression g^ + ^+a^ W 

which, as we have seen, measures the rate of increase of volume of the fluid 
at the point (a?, y, 2), is conveniently called the ^expansion' at that point. 
From a more general point of view the expression (4) is called the 
* divergence' of the vector (u, v, w); it is often denoted briefly by 

div (w, V, w). 

The preceding investigation is substantially that given by Euler*. 
Another, and now more usual, method of obtaining the equation of 
continuity is, instead of following the motion of a fluid element, to fix the 
attention on an element 8x8ySz of space, and to calculate the change produced 
in the included mass by the flux across the boundary. If the centre' of the 
element be at {x, y, z), the amount of matter which per unit time enters it 
across the y2;-face nearest the origin is 

and the amount which leaves it by the opposite face is 



(■ 



/>w + i - g|^ Bxj 8y8z. 



* 2.C. ante p. 2. 



6 The Equations of Motion [chap, i 

The two faces together give a gain 

per unit time. Calculating in the same way the effect of the flux across the 
remaining faces, we have for the total gain of mass, per unit time, in the 
space hxhyhz, the formula 

Since the quantity of matter in any region can vary only in consequence 
of the flux across the boimdary, this must be equal to 

-^^{phxhyhz), 

whence we get the equation of continuity in the form 

a< "*" ~a^ + " ajT "^ "ar - *^ ^^^ 

8. It remains to put in evidence the physical properties of the fluid, so 
far as these affect the quantities which occur in our equations. 

In an 'incompressible' fluid, or liquid, we have Dp/Dt == 0, in which case 
the equation of continuity takes the simple form 

s^-i-i-"- <■' 

It is not assumed here that the fluid is of uniform density, though this is 
of course by far the most important case. 

If we wished to take account of the slight compressibility of actual liquids, 
we should have a relation of the form 

p = K{p- po)/po. • • • (2) 

or pIPo^I + P/k, (3) 

where k denotes what is called the 'elasticity of volume.' 

In the case of a gas whose temperature is uniform and constant we have 
the 'isothermal' relation 

pIpo^pIpo W 

where p^, pQ are any pair of corresponding values for the temperature in 
question. 

In most cases of motion of gases, however, the temperature is not constant, 
but rises and falls, for each element, as the gas is compressed or rarefied. 
When the changes are so rapid that we can ignore the gain or loss of heat 
by an element due to conduction and radiation, we have the 'adiabatic' 
relation 

pIPo = (p/Po)^y (5) 



7-9] Boundary Condition 7 

where Pq and p^ are any pair of corresponding values for the element con- 
sidered. The constant y is the ratio of the two specific heats of the gas ; 
for atmospheric air, and some other gases, its value is 1*408. 

9. At the boundaries (if any) of the fluid, the equation of continuity 
is replaced by a special surface-condition. Thus at a fixed boundary, the 
velocity of the fluid perpendicular to the surface must be zero, i.e. if I, m, n 
be the direction-cosines of the normal, 

lu-i- mv -{- nw = (1) 

Again at a surface of discontinuity, i.e. a surface at which the values of 
Uf V, w change abruptly as we pass from one side to the other, we must have 

I (wi — w,) -f- m (vi — Vj) + n (t<^i — w;^) = 0, (2) 

where the suffixes are used to distinguish the values on the two sides. 
The same relation must hold at the common surface of a fluid and a moving 
soUd. 

The general surface-condition, of which these are particular cases, is that 

if F {x^ y, z, ^) = be the equation of a boimding surface, we must have at 

every point of it 

DFIDt = Q (3) 

For the velocity relative to the surface of a particle lying in it must be 
wholly tangential (or zero), otherwise we should have a finite flow of fluid 
across it. It follows that the instantaneous rate of variation of F for a 
surface-particle must be zero. 

A fuller proof, given by Lord Kelvin*, is as follows. To find the rate 
of motion (v) of the surface F (x, y, 2, t) = 0, normal to itself, we write 

F(x-\- IvU, y + mvU, z + nvht, « + SO = 0, 

where Z, m, n are the direction-cosines of the normal at (x, y, 2), whence 



.( 



,dF ^ dF^ dF\dF - 



Since (I,m.«) = (^. ^. -g- j - B, 

we have ^ '^ "" »"Sf" ^^^ 

At every point of the surface we must have 

v = lu + mv -\- nw, 
which leads, on substitution of the above values of {, m, n, to the equation (3). 

* (W. Thomaon) "Notes on Hydrodynamios,*' Comb, and Dub. Ifaih. Joum. Feb. 1848. 
[MaOiemiUical and Physical Papers, Cambridge, 1882..., t. i. p. 83.] 



8 The EqtMtions of Motion [chap, i 

The partial differential equation (3) is also satisfied by any surface 
moving with the fluid. This follows at once from the meaning of the operator 
DjDt, A question arises as to whether the converse necessarily holds; i.e. 
whether a moving surface whose equation -F = satisfies (3) will always 
consist of the same particles. Considering any such surface, let us fix our 
attention on a particle P situate on it at time t. The equation (3) expresses 
that the rate at which P is separating from the surface is at this instant 
zero ; and it is easily seen that if the motion be oontinv/ous (according to the 
definition of Art. 4), the normal velocity, relative to the moving surface -F, 
of a particle at an infinitesimal distance t, from it is of the order ^, viz. it is 
equal to Gt, where G is finite. Hence the equation of motion of the particle 
P relative to' the surface mav be written 

Dtim = Gi 

This shews that log t, increases at a finite rate, and since it is negative infinite 
to begin with (when t, = 0), it remains so throughout, i.e. f remains zero for 
the particle P. 

The same result follows from the nature of the solution of 

dF dF dF dF ^ 

considered as a partial differential equation in ^*. The subsidiary system of ordinary 
differential equations is 

dt=^=^=^ (6) 

in which x, y, z are regarded as functions of the independent variable t. These are 
evidently the equations to find the paths of the particles, and their integrals may be 
supposed put in the forms 

x=fi (a, b, c, t), y=f^ (a, 6, c, t), z=U(a,h,c,t\ (7) 

where the arbitrary constants a, 6, c are any three quantities serving to identify a particle ; 
for instance they may be the initial co-ordinates. The general solution of (5) is then found 
by elimination of a, h, c between (7) and 

JP = Vr (a, 6, c) (8) 

where ^ is an arbitrary function. This shews that a particle once. in the surface F=0 
remains in it throughout the motion. 



Eqiuxtion of Energy, 

10. In most cases which we shall have occasion to consider the extraneous 
forces have a potential ; viz. we have 

an an an 

* Lagrange, Oeuvres, t. iv. p. 706. 



9-10] Energy 9 

The physical meaning of fl is that it denotes the potential energy, per unit 
mass, at the point (2;, y, z), in respect of forces acting at a distance. It will 
be sufficient for the present to consider the case where the field of extraneous 
force is constant with respect to the time, i,e. dO./dt = 0. If we now multiply 
the equations (2) of Art. 6 by u, v, w, in order, and add, we obtain a result 
which may be written 

If we multiply this by hxhyhz^ and integrate over any region, we find 

|(r+F).-///(.| + .| + »|)^*^ (2, 

where T = J///p (w« + i;« -{- 1(;«) (fodydz, V = Siiilpdzdydz, ...(3) 

i.e. T and V denote the kinetic energy, and the potential energy in relation 
to the field of extraneous force, of the fluid which at the moment occupies 
the region in question. The triple integral on the right-hand side of (2) 
may be transformed by a process which will often recur in our subject. Thus, 
by a partial integration, 

jjj af ^^^y^^ ^ jj^^ ^y^^ "■ jjjP gix ^^^y^^> 

where [pu\ is used to indicate that the values of jni at the points where the 
boimdary of the region is met by a line parallel to x are to be taken, with 
proper signs. If Z, m, n be the direction-cosines of the inwardly directed 
normal to any element SjS of this boundary, we have hyhz — ± 188, the signs 
alternating at the successive intersections referred to. We thus find that 

!S[jm]dydz = - J/pw IdS, 

where the integration extends over the whole boimding surface. Trans- 
forming the remaining terms in a similar manner, we obtain 

^{T + V)=jfp{lu + mv + nw)dS + jjjp(^+^ + ^')dxdydz...{4:) 
In the case of an incompressible fluid this reduces to the form 

^1 S^^ "*" ^^ " f/^'^ + mv 4- nw)pdS (5) 

Since Zw -f- mt? -h nw denotes the velocity of a fluid particle in the direction of 
the normal, the latter integral expresses the rate at which the pressures fhS 
exerted from without on the various elements hS of the boundary are doing 
work. Hence the total increase of energy, kinetic and potential, of any 



10 The Equations of Motion [chap, i 

portion of the liquid, is equal to the work done by the pressures on its 
surface. 

In particular, if the fluid be bounded on all sides by fixed waUs, we have 

over the boundary, and therefore 

T 4- F = const (6) 

A similar interpretation can be given to the more general equation (4), 
provided p be a function of p only. If we write 



--Hi) i'» 



E 

then E measures the work done by unit mass of the fluid against external 
pressure, as it passes, under the supposed relation between p and p, from its 
actual volume to some standard volume. For example, if the unit mass 
were enclosed in a cylinder with a sliding piston of area A, then when the 
piston is pushed outwards through a space 8x, the work done is pA . Sx, of 
which the factor ASx denotes the increment of volume, i.e, of p"^. In the 
case of the adiabatic relation we find 



^ = -1^(2-20) (8) 

We may call E the intrinsic energy of the fluid, per unit mass. Now, 
recalling the interpretation of the expression 

du dv dw 
dx dy dz 

given in Art. 7, we see that the volimie-integral in (4) measures the rate at 
which the various elements of the fluid are losing intrinsic energy by 
expansion*; it is therefore equal to — DW/Dt, 

where W = /// Epdxdydz (9) 

Hence (^t -{- V + W) ^jjp {lu 4- mv + nw)dS (10) 

• 
The total energy, which is now partly kinetic, partly potential in relation to 
a constant field of force, and partly intrinsic, is therefore increasing at a rate 
equal to that at which work is being done on the boundary by pressure from 
without. 



10-11] 



ImpuUive Motion 



11 



Impulsive Generation of Motion. 

11. If at any instant impulsive forces act bodily on the fluid, or if the 
boundary conditions suddenly change, a sudden alteration in the motion may 
take place. The latter case may arise, for instance, when a soUd immersed 
in the fluid is suddenly set in motion. 

Let p be the density, u, t;, w the component velocities immediately before, 
u', v\ w' those immediately after the impulse, X\ Y\ Z' the components of 
the extraneous impulsive forces per unit mass, m the impulsive pressure, at 
the point (x, y, z). The change of momentum parallel to x of the element 
defined in Art. 6 is then pSxSySz i^' — vl) ; the x-component of the extraneous 
impulsive forces is p8:s8ySz X\ and the resultant impulsive pressure in the 
same direction is — 'dmfdx . hxhyhz. Since an impulse is to be regarded as an 
infinitely great force acting for an infinitely short time (r, say), the effects of 
all finite forces during this interval are neglected. 

Hence, pSxSySz {u' — w) = pSxSySz-X' — -^ &r%8z, 



or 



Similarly, 



, «., 13tD 

u — w=A — ^— . 

pox 

pdy 

W — W = L -^- , 

p oz 



> 



(1) 



These equations might also have been deduced from (2) of Art. 6, by 
multiplying the latter by 8^, integrating between the limits and r, putting 

X'^Txdt, Y'^Trdt, Z'^Tzdt, w^Wdt, 
JO Jo Jo Jo 

and then ma]dng r vanish. 

In a Uquid an instantaneous change of motion can be produced by the 
action of impulsive pressures only, even when no impulsive forces act bodily 
on the mass. In this case we have X\ Y\ Z' = 0, so that 



, 13tD 

u — w = — ^5— 

p ox 

, 13oj . 

pdy ^ 

, I dm 

p oz \ 



(2) 



If we differentiate these equations with respect to x, y, 2;, respectively, and 



12 The Bqtiations of Motion [chap, i 

add, and if we further suppose the density to be uniform, we find by Art. 8(1) 

that 

d^w d^w d^rn 
4- 4- = 

The problem then, in any given case, is to determine a value of m satisfying 
this equation and the proper boundary conditions* ; the instantaneous change 
of motion is then given by (2). 



Equations referred to Moving Axes. 

12. It is sometimes convenient in special problems to employ a system 
of rectangular axes which is itself in motion. The motion of this frame may 
be specified by the component velocities n, v, w of the origin, and the com- 
ponent rotations p, q, r, all referred to the instantaneous positions of the 
axes. If u, V, w be the component velocities of a fluid particle at (x^ y, z), 
the rates of change of its co-ordinates relative to the moving frame will be 

^ = u-u + Ty-qz, -^ = t;-v + pz-rx, ^^w-w + qx-i^y, . ,(l) 

After a time ht the velocities of the particle parallel to the new positions 
of the co-ordinate axes will have become 

To find the component accelerations we must resolve these parallel to 
the original positions of the axes in the manner explained in books on 
Dynamics. In this way we obtain the expressions 

du duDx duDy duDz ' 

dv dv Dx , dv Dy , dv Dz \ /«x 

dt-^+'^+diDt + TylA + diDf^ (^) 

dw dwDx dtoDy dwDz 

_ ^Qw + pv +__ + — _+ ^ — .^ 

These will replace the expressions in the left-hand members of Art. 6 (2)'|'. 
The general equation of continuity is 

hU'^Ki^^hU^^h" <*) 

* It will appear in Chapter m. that the value of w is thus detemuDate, save as to an 
additive constant. 

t GreenhiU, "On the General Motion of a Liquid EUipsoid. . .," Proc, Camb. PhU. Soe. t. iv. 
p. 4 (1880). 



11-14] Lagrangian Eqvxztions 13 

reducing in the case of incompressibility to the form 

du dv dw 

di^d-y^Tz^^ (^) 

as before. 

The Lagrangian Equations. 

13. Let a, 6, c be the initial co-ordinates of any particle of fluid, x, y, z 
its co-ordinates at time L We here consider x, y, z bls functions of the 
independent variables a, b^ Cyt; their values in terms of these quantities give 
the whole history of every particle of the fluid. The velocities parallel to 
the axes of co-ordinates of the particle (a, b, c) at time t are dx/dt, dy/dt, dz/dt, 
and the component accelerations in the same directions are dh>/dt\ dhf/dt\ 
dh/dfi. Let p be the pressure and p the density in the neighbourhood of 
this particle at time t; X, Y, Z the components of the extraneous forces per 
imit mass acting there. Considering the motion of the mass of fluid which 
at time t occupies the differential element of volume Sxiyhz, we find, by the 
same reasoning as in Art. 6, 





X- 


Idp 

pdx' 




Y- 


Idp 




Z- 


Idp 
pdz' 



These equations contain differential coefficients with respect to x, y, z, 
whereas onr independent variables are a, b, c, t. To eliminate these dif- 
ferential coefficients, we multiply the above equations by dxjda, dy/da, dzjda, 
respectively, and add ; a second time by dxjdb, dy/db, dz/db, and add ; and again 
a third time by dxfdc, dyjdc, dz/dc, and add. We thus get the three equations 

\dt* "^Jda^Kdt* ^)da'^\dt* ^Jda'^pda'^' 

\dt* "^Jdb^Kdt* )db^\dt*~^)db^ pdb~^' 

'd*x_ \dx (dhi y\3.y,/a»e \dz idp_ 
\di* ^)d'c'^\dr*~^)dE^W-^)d'c'^~pd'c-^- 

These are the 'Lagrangian' forms of the dynamical equations. 

14. To find the form which the equation of continuity assumes in terms 
of our present variables, we consider the element of fluid which originally 
occupied a rectangular parallelepiped having its centre at the point (a, 6, c), 
and its edges 8a, S6, 8c parallel to the axes. At the time t the same element 
forms an oblique parallelepiped. The centre now has for its co-ordinates 



14 



The Equations of Motion 



[chap. I 



X, y, z; and the projections of the edges on the co-ordinate axes are 
respectively 

3^^' i^'*' d^^' 

36^*' 06^' db^^' 
dx ^, dy ^ 3z a 

;^~ OC» ^ OCm x~ oc» 

oc oc oc 

The volume of the parallelepiped is therefore 

dx dy dz 

3a ' da' da 

dx dy 
db' 36' 



dx 



dy 

dc* 



dj_ 

db 

dz^ 
dc 



8ahbSc, 



or, as it is often written, 



(a, 6, c) 



Hence, since the mass of the element is unchanged, we have 

^ 3 (.r, y, z ) 

^dWhTcr^'^ 

where />o is the initial density at (a, 6, c). 

In the case of an incompressible fluid p = Po, ^o that (1) becomes 

3 (x, y, z) 



(1) 



3 (a, 6, c) 



= 1. 



(2) 



Weber's Transformation. 

15. If as in Art. 10 the forces X, Y, Z have a potential H, the dynamical 
equations of Art. 13 may be written 

d^dx d*ydy dhdz 3a_13p . , 

dt^ da ^ dt^ da ^ dt^ da~ da pda' *^" *^- 

Let us integrate these equations with respect to t between the limits and L 
We remark that 



I 



f*d^dx^^ r3x3x"]<_ f^dx 
Jodt^da "[dtdajo Jodt 



d*x 



dt ' 



__dxdx - 3 [^ fdx\* ^ 

"dtd^^'^'^^^diJoKdi) ""' 



14-16] Weber's Transformation 15 

where t^o is the initial value of the x-component of velocity of the particle 
(a, 6, c). Hence if we write 



we find* 



-/:[/?^"-M(S)'^(IA©T 



dx dx dy dy dzdz _ dx. 



*. (1) 



^ax ayay^^_ __ax.i /ov 

a<a6'^a<a6'*'a<a6 *"~ a6''" ^^ 

dxdx dydy dzdz _ a^ 

a< ac "^ ^ac ■*■ a< ac ~ "'» ~ ~ ac • / 

These three equations, together with 

i-/?-"-*{(i)'-(i)'-(ir} (') 

and the equation of continuity, are the partial differential equations to be 
satiBfied by the five unknown quantities x, y, z, p, x'y P being supposed 
already eliminated by means of one of the relations of Art. 8. 

The initial conditions to be satisfied are 

16. It is to be remarked that the quantities a, 6, c need not be restricted 
to mean the initial co-ordinates of a particle ; they may be any three quanti- 
ties which serve to identify a particle, and which vary continuously from one 
particle to another. If we thus generaUze the meanings of a, 6, c, the form 
of the dynamical equations of Art. 13 is not altered ; to find the form which 
the equation of continuity assumes, let Xq, y^, z^ now denote the initial 
co-ordinates of the particle to which a, 6, c refer. The initial volume of the 
parallelepiped, whose centre is at (xq, y^, Zq) and whose edges correspond to 
variations 8a, 86, 8c of the parameters, a, 6, c, is 

a (a, 6, c) 

so that we nave p ^. — j^ — r = po -5-7 — . x , (1) 

"^ d (a, 6, c) '^" d (a, 6, c) ' y 

or, for an incompressible fluid, 

8 (a?> y, g) ^ 3 (a? o>yo, gp) ^2) 

9 (a, 6, c) 3 (a, 6, c) ^ ' 

* H. Weber, **Ueber eine Traoaformation der hydiodyiianuBoheii Gleichungen," Crelle, 
t. Ixviii (1868). It is assumed in (1) that the density p, if not uniform, is a function of p only. 



CHAPTER II 

INTEGRATION OF THE EQUATIONS IN SPECIAL CASES 

17. In a large and important class of cases the component velocities 
tiy V, w can be expressed in terms of a single function <f>, as follows : 

Such a function is called a 'velocity-potential/ from its analogy with the 
potential function which occurs in the theories of Attractions, Electro- 
statics, &c. The general theory of the velocity-potential is reserved for the 
next chapter; but we give at once a proof of thie following important 
theorem : 

If a velocity-potential exist, at any one instant, for any finite portion of 
a perfect fluid in motion under the action of forces which have a potential, 
then, provided the density of the fluid be either constant or a function of the 
pressure only, a velocity-potential exists for the same portion of the fluid at 
all instants before or after f. 

In the equations of Art. 15, let the instant at which the velocity- 
potential <f>Q exists be taken as the origin of time; we have then 

^ 4 

throughout the portion of the mass in question. Multiplying the equa- 
tions (2) of Art. 15 in order by (Ja, 3h^ dc, and adding, we get 

^ (fo -f P^ rfy + gy rf* - {u^da 4- v^db -f t^o*^) = — dx, 

s 

* The reasons for the introduction of the mtniM sign are stated in the Preface. 

t Lagrange, "M^moire sur la Thtorie dii Mouvement des Eluides," Nouv. mim. de VAcad, de 
Berlin, 1781 [Oeuvrea, t. iv. p. 714]. The argument is reproduced in the Micanique Analytique, 

Lagrange's statement and proof were alike imperfect; the first rigorous demonstration is due 
to Caucby, **M6moire sur la Throne des Ondes," M^. de VAcad. roy. des Sciences, t. i. (1827) 
[Oeuvres Computes, Paris, 1882 ..,,V* S^rie, t. i. p. 38] ; the date of the memoir is 1815. Another 
proof is given by Stokes, Cawb. Trans, t. viii. (1845) (see also Math, and Pkys. Papers, Cam- 
bridge, 1880. . ., t. i pp. 106, 158, and t. ii. p. 36), together with an exceUent historical and 
critical account of the whole matter. 



17^18] Velocity-Potential 17 

or, in tha'Eulerian' notation, 

udx + vdy + wdz = — <? (^o + x) ~ "~ ^» ^^7' 
Since the upper limit of t in Art. 15 (1) may be positive or negative, this 
proves the theorem. 

It is to be particularly noticed that this continued existence of a velocity- 
potential is predicated, not of regions of space, but of portions of matter. 
A portion of matter for which a velocity-potential exists moves about and 
carries this property with it, but the part of space which it originally occupied 
may, in the course of time, come to be occupied by matter which did not 
originally possess the property, and which therefore cannot have acquired it. 

The class of cases in which a velocity-potential exists includes all those 
where the motion has originated from rest under the action of forces of the 
kind here supposed ; for then we have, initially, 

u^da + v^db + '^Jb^dc = 0, 
or ff>Q = const. 

The restrictions under which the above theorem has been proved must 
be carefully remembered. It is assumed not only that the extraneous forces 
X, y , Z, estimated at per unit mass, have a potential, but that the density p 
is either uniform or a function of p only. The latter condition is violated, 
for example, in the case of the convection currents generated by the unequal 
appUcation of heat to a fluid; and again, in the wave-motion of a hetero- 
geneous but incompressible fluid arranged originally in horizontal layers 
of equal density. Another case of exception is that of 'electro-magnetic 
rotations'; see Art. 29. 

18. A comparison of the formulae (1) with the equations (2) of Art. 11 
leads to a simple physical interpretation of <f>. 

Anj actual state of motion of a liquid, for which a (single-valued) 
velocity-potential exists, could be produced instantaneously from rest by the 
appUcation of a properly chosen system of impulsive pressures. This is evident 
from the equations cited, which shew, moreover, that <^ = mjp + const. ; so 
that tn = fxf> -\- C gives the requisite system. In the same way w = — p<f> -\- 
gives the system of impulsive pressures which would completely stop the 
motion. The occurrence of an arbitrary constant in these expressions shews, 
what is otherwise evident, that a pressure uniform throughout a liquid mass 
produces no effect on this motion*. 

In the case of a gas, <f> may be interpreted as the potential of the extraneous 
impulsive forces by which the actual motion at any instant could be produced 
instantaneously from rest. 

* This interpretation was given by Cauohy, loc, cit., and by Poisson, Mim. deVAcad. roy. 
des Sciences, t. i. (1816). 

L.H. 2 



18 Integration of the Eqvxztions in Special Cases [chap, ii 

A state of motion for which a velocity-potential does not exist cannot be 
generated or destroyed by the action of impulsive pressures, or of extraneous 
impulsive forces having a potential. 

19. The existence of a velocity-potential indicates, besides, certain 
Hnemalical properties of the motion. 

A *line of motion' or * stream-line'* is defined to be a hne drawn from 
point to point, so that its direction is everywhere that of the motion of the 
fluid. The differential equations of the system of such lines are 

The relations (1) shew that when a velocity-potential exists the lines of 
motion are everywhere perpendicular to a system of surfaces, viz. the 
' equipotentiar surfaces ff> = const. 

Again, if from the point (x, y, z) we draw a linear element hs in the 
direction (Z, w, w), the velocity resolved in this direction \&lu-\- mv •\- nw, or 

d<f> dx d<f>dy dif>dz i,- i, _ ^^ 

The velocity in any direction is therefore equal to the rate of decrease of 
<f> in that direction. 

Taking hs in the direction of the normal to the surface (f> = const., we see 
that if a series of such surfaces be drawn corresponding to equidistant values 
of <f}y the common difference being infinitely small, the velocity at any point 
will be inversely proportional to the distance between two consecutive surfaces 
in the neighbourhood of the point. 

Hence, if any equipotential surface intersect itself, the velocity is zero 
at the intersection. The intersection of two distinct equipotential surfaces 
would imply an infinite velocity. 

20. Under the circumstances stated in Art. 17, the equations of motion 
are at once integrable throughout that portion of the fluid mass for which 
a velocity-potential exists. For in virtue of the relations 

dv _dw dw ^ du du __ dv 
dz^ dy* dx~ dz' dy^dx* 

which are implied in (1), the equations of Art. 6 may be written 

dxdi dx dx dx "^ dx pdx* '' 

* Some writers prefer to restrict the use of the term * stream-line' to the case of steady 
motion, as defined in Art 21. 



18-21] Vdodty-Potmtial 19 

These have the integral 



/' 



■f-i-a-J«' + m (3) 

where q denotes the resultant velocity (w* + t;* + y^) j and F(t) is an arbitrary 
function of t. 



Our equations take a specially simple form in the case of an incompressible 
fluid ; viz. we then have 

l^^-n-\f + F(t), (4) 

with the equation of continuity 

a^'^a^'^a?^^' ^^) 

which is the equivalent of Art. 8 (1). When, as in many cases which we 
shall have to consider, the boundary conditions are purely kinematical, the 
process of solution consists in finding a function which shall satisfy (5) and v 

the prescribed surface-conditions. The pressure p is then given by (4), and 
is thus far indeterminate to the extent of an additive function of t. It 
becomes determinate when the value of p at some point of the fluid is given 
for all values of t. Since the term F (t) is without influence on resultant 
pressures it is frequently omitted. 

Suppose, for example, that we have a solid or solids moving through a liquid com- 
pletely enclosed by fixed boundaries, and that it is possible (t.g, by means of a piston) to 
apply an arbitrary pressure at some point of the boundary. Whatever variations are made 
in the magnitude of the force applied to the piston, the motion of the fluid and of the 
solids will be absolutely unaffected, the pressure at all points instantaneously rising or 
falling by equal amounts. Physically, the origin of the paradox (such as it is) is that the 
fluid is treated as absolutely incompressible. In actual liquids changes of pressure are 
propagated with very great, but not infinite, velocity. 

K the co-ordinate axes are in motion, the formula for the pressure is 

-'(^l-|)-«('l-l)-'('|-'g). ■■•■'«' 

where g* = (w - ii)^ + (v - v)* ^ (w - w)^ (7) 

This easily follows from the formulae for the accelerations given in Art. 12 (3). 

Steady Motion. 

21. When at every point the velocity is constant in magnitude and 
direction, i.e. when 

dt ^' dt "' dt ^' ^^^ 

everywhere, the motion is said to be * steady.' 

2—2 



20 Integration of the Equations in Special Cases [chap, ii 

In steady motion the lines of motion coincide with the paths of the 
particles. For if P, Q be two consecutive points on a line of motion, 
a particle which is at any instant at P is moving in the direction of the 
tangent at P, and will, therefore, after an infinitely short time arrive at Q, 
The motion being steady, the lines of motion remain the same. Hence the 
direction of motion at Q is along the tangent to the same line of motion, 
i.e, the particle continues to describe the line. 

In steady motion the equation (3) of the last Article becomes 

dp 






= - II - iflr2 -i- constant (2) 

P 



The law of variation of pressure along a stream-line can however in this case 
be found without assuming the existence of a velocity-potential. For if hs 
denote an element of a stream-line, the acceleration in the direction of 
motion is qdq/dSy and we have 

dq ^ 9ft 1 dp 
^ 3« "" ds pds' 

whence, integrating along the stream-line, 

'dp 



I- 



= -ft-igr« + C (3) 



This is similar in form to (2), but is more general in that it does not assume 
the existence of a velocity-potential. It must however be carefully noticed 
that the 'constant' of equation (2) and the 'C of equation (3) have different 
meanings, the former being an absolute constant, while the latter is constant 
along any particular stream-line, but may vary as we pass from one stream- 
line to another. 

22. The theorem (3) stands in close relation to the principle of energy. 
If this be assumed independently, the formula may be deduced as follows*. 
Taking first the particular case of a hquid, let us consider the portion of an 
infinitely narrow tube, whose boundary follows the stream-hnes, included 
between the cross-sections A and J5, the direction of motion being from A 
to J5. Let p be the pressure, q the velocity, ft the potential of the extraneous 
forces, a the area of the cross-section, at A, and let the values of the same 
quantities at B be distinguished by accents. In each unit of time a mass 
pqa at A enters the portion of the tube considered, whilst an equal mass 
pq'a leaves it at J5. Hence qa = qW, Again, the work done on the mass 
entering at ^ is pqa per unit time, whilst the loss of work at J3 is p'((o , 
The former mass brings with it the energy pqa {\(^ + ft), whilst the latter 
carries off energy to the amount pjV (Jg'* + ft'). The motion being steady, 

* This is really a reversion to the methods of Daniel Bernoulli, Hydrodynamica, Argentorati, 
1738. 



21-23] Steady Motion 21 

the portion of the tube considered neither gains nor loses energy on the 
whole, so that 

Dividing by pqa (= pq'a), we have 

2 + i?^ + n = ^ + i?'* + n', 
p p 

or, using C in the same sense as before, 

J=-ll-iy« + C, (4) 

which is what the equation (3) becomes when p is constant. 

To prove the corresponding formula for compressible fluids, we remark 
that the fluid entering at A now brings with it, in addition to its energies 
of motion and position, the intrinsic energy 

per tmit mass. The addition of these terms to (4) gives the equation (3). 
The motion of a gas is as a rule subject to the adiabatic law 

P/Po = (p//>o)^ (5) 

and the equation (3) then takes the form 

^jJ = -ft-k« + C (6) 

23. The preceding equations shew that, in steady motion, and for points 
along any one stream-line *, the pressure is, cceteris paribus, greatest where 
the velocity is least, and vice versa. This statement, though opposed to 
popular notions, becomes evident when we reflect that a particle passing 
from a place of higher to one of lower pressure must have its motion 
accelerated, and vice V€rsd'\. 

It follows that in any case to which the equations of the last Article 
apply there is a limit which the velocity cannot exceed f. For instance, let 
us suppose that we have a liquid flowing from a reservoir where the velocity 
may be neglected, and the pressure is ^q, and that we may neglect extraneous 
forces. We have then, in (4), C = pp/p, and therefore 

P = J>o-iP?' (7) 

Now although it is found that a Uquid from which all traces of air or other 
dissolved gas have been eliminated can sustain a negative pressure, or tension, 

* This restriction is mmeceBsary when a velooity-potential exists. 

t Some interesting practical illustrations of this principle are given by Fronde, Nature, 
t. xiii 1875. 

% Cf. Helmholtz, **Ueber discontinuirliche flassigkeitsbewegnngen/' BerL MonaJUber. April 
1868; Pha, Mag, Not. 1808 [WisaenachafiUcke Abhandlungen, Leipzig, 1882-3, t. i. p. 146]. 



22 Integration of the Equations in Special Coms [chap, n 

of considerable magnitude*, this is not the case with fluids such as we find 
them under ordinary conditions. Practically, then, the equation (7) shews 

that g cannot exceed (2po/p) • This limiting velocity is of course that with 
whicb the fluid would escape from the reservoir into a vacuum. In the case 
of water at atmospheric pressure it is the velocity 'due to' the height of the 
water-barometer, or about 45 feet per second. 

If in any case of fluid motion of which we have succeeded in obtaining 
the analytical expression, we suppose the motion to be gradually accelerated 
until the velocity at some point reaches the limit here indicated, a cavity will 
be formed there, and the conditions of the problem are more or less changed. 

It will be shewn, in the next chapter (Art. 44), that in irrotational motion 
of a liquid, whether 'steady' or not, the place of least pressure is always at 
some point of the boundary, provided the extraneous forces have a potential fl 
satisfying the equation 

This includes, of course, the case of gravity. 

In the general case of a fluid in which ;> is a given function of p we have, 
putting fl = in (3), 



=2r^ (8) 

J» p 



V 9 



For a gas subject to the adiabatic law, this gives 

if c, = iyp/pr, = {dp/dpy, denote the velocity of sound in the gas when at 
pressure p and density />, and Cq the corresponding velocity for gas under the 
conditions which obtain in the reservoir. (See Chapter x.) Hence the limiting 

velocity is 

2_\i 

or2-214co, ify=l-408. 

24. We conclude this chapter with a few simple appUcations of the 
equations. 

Effiux of Liquids. 

Let us take in the first instance the problem of the efflux of a Uquid from 
a small orifice in the walls of a vessel which is kept filled up to a constant 
level, so that the motion may be regarded as steady. 

♦ O. Reynolds, Manch. Mem. t. vL (1877) [Scieniijic Papers, Cambridge, 1900. . ., t. L p. 231]. 



(, 



•^o> 



23-24] Efflux of Liquids 23 

The origin being taken in the upper surface, let the axis of z be vertical, 
and its positive direction downwards, so that il= — gz. If we suppose the 
area of the upper surface large compared with that of the orifice, the velocity 
at the former may be neglected. Hence, determining the value of C in 
Art. 22 (4) so that p^ P (the atmospheric pressure), when 2 = 0, we have* 

l^l + gz-^q» (1) 

r r 

At the surface of the issuing jet we have p ^^ P, and therefore 

?*=2<^, (2) 

i,e, the velocity is that due to the depth below the upper surface. This is 
known as TorriceMs Theorem^, 

We cannot however at once apply this result to calculate the rate of efflux 
of the fluid, for two reasons. In the first place, the issuing fluid must be 
regarded as made up of a great number of elementary streams converging 
from all sides towards the orifice. Its motion is not, therefore, throughout 
the area of the orifice, everywhere perpendicular to this area, but becomes 
more and more oblique as we pass from the centre to the sides. Again, the 
converging motion of the elementary streams must make the pressure at the 
orifice somewhat greater in the interior of the jet than at the surface, where 
it is equal to the atmospheric pressure. The velocity, therefore, in the interior 
of the jet will be somewhat less than that given by (2)*. 

Experiment shews however that the converging- motion above spoken of 
ceases at a short distance beyond the orifice, and that (in the case of a circular 
orifice) the jet then becomes approximately cylindrical. The ratio of the area 
of the section S' of the jet at this point (called the 'vena contracta') to tbe 
area S of the orifice is called the 'coefficient of contraction.' If the orifice be 
simply a hole in a thin wall, this coefficient is found experimentally to be 
about -62. 

The paths of the particles at the vena contracta being nearly straight, 
there is little or no variation of pressure as we pass from the axis to the outer 
surface of the jet. We may therefore assume the velocity there to be imiform 
throughout the section, and to have the value given by (2), where z now 
denotes the depth of the vena contracta below the surface of the Uquid in the 
vessel. The rate of efflux is therefore 

{^z)^-pS' (3) 

The calculation of the form of the issuing jet presents difficulties which 
have only been overcome in a few ideal cases of motion in two dimensions. 
(See Chapter iv.) It may however be shewn that the coefficient of con- 
traction must, in general, he between J and 1. To put the argument in its 

* ThiB result is due to D. Bernoulli, 2.c. ante p. 20. 

t '*I>e motu grayium naturaliter accelerato," Firenze, 1643. 



24 Integration of the Equations in Special Cases [chap, ii 

simplest form, let us first take the case of liquid issuing from a vessel the 
pressure in which, at a distance from the orifice, exceeds that of the external 
space by the amount P, gravity being neglected. When the orifice is closed 
by a plate, the resultant pressure of the fluid on the containing vessel is of 
course nil. If when the plate is removed we assume (for the moment) that 
the pressure on the walls remains sensibly equal to P, there will be an un- 
balanced pressure PS acting on the vessel in the direction opposite to that of 
the jet, and tending to make it recoil. The equal and contrary reaction on 
the fluid produces in unit time the velocity q in the mass pqS' flowing through 
the *vena contracta,' whence 

PS = pqK^' (4) 

The principle of energy gives, as in Art. 22, 

P = W, (5) 

SO that, comparing, we have S' = ^S, The formula (1) shews that the 
pressure on the walls, especially in the neighbourhood of the orifice, will in 
reality fall somewhat below the static pressure P, so that the left-hand side 
of (4) is an under-estimate. The ratio S'/S will therefore in general be > J. 

In one particular case, viz. where a short cylindrical tube, projecting 
inwards, is attached to the orifice, the assumption above made is sufficiently 
exact, and the consequent value | for the coefficient then agrees with 
experiment. 

« 

The reasoning is easily modified so as to take account of gravity (or other 
conservative forces). We have only to substitute for P the excess of the static 
pressure at the level of the orifice over the pressure outside. The difference 
of level between the orifice and the 'vena contracta' is here neglected*. 

Efflux of Gases. 

25. We consider next the efflux of a gas, supposed to flow through a 
small orifice from a vessel in which the pressure is p^ and density p^ into a 
space where the pressure is pi. We assume that the motion has become 
steady, and that the expansion takes place according to the adiabatic law. 

If the ratio pjpi of the pressures inside and outside the vessel do not exceed a certain 
limit, to he indicated presently, the flow will take place in much the same manner as in 
the case of a liquid, and the rate of discharge may be found by putting p =pi in Art. 23 (9), 

♦ The above theory is due to Borda {Mim. de VAcad. des Sciences, 1766), who also made 
experiments with the special form of mouth-piece referred to, and found 818' = 1-942. It was 
re-discovered by Hanlon, Proa, Lond, Maih, 8oc, t. iii p. 4 (1869); the question is further 
elucidated in a note appended to this paper by MaxwelL See also Froude and J. Thomson, 
Proc, Olasgoto Phil. Soc. t. x. (1876). It has been remarked by several writers that in the case 
of a diverging conical mouth-piece projecting inwards the section at the vena contracta may be 
less than half the area of the internal orifice. 



24-26] Efflux of Oases 25 

and multipl3diig the resulting value of q by the area 8' of the vena oontraota. This gives 
for the rate of discharge of mass * 

2 y+1 

,.««'-C4,)'..{(g)'-© ')'•»■■ <■' 

It is plain, however, that there must be a limit to the applicability of this result ; for 
otherwise we should be led to the paradoxical conclusion that when py^ =0, i.e. the discharge 
is into a vacuum, the flux of matter is nil. The elucidation of this point is due to 
Prof. Osborne Reynolds f- It is easily found by means of Art. 23 (8), that ^p is a maximum, 
i.e. the section of an elementary stream is a minimum, when g* = dp/dp, that is, the velocity 
of the stream is equal to the velocity of sound in gas of the pressure and density which 
prevaO there. On the adiabatic hypothesis this gives, bv Art. 23 ( 10), 

HM'-- «' 

and therefore, since c* qc p^ , 

H^f- ^(4l)'"^ '" 

or, if y = 1-408, p = -634po. ^ = -527po (4) 

If Pi be less than this value, the stream -after passing the point in question widens out 
again, until it is lost at a distance in the eddies due to viscosity. The minimum sections 
of the elementary streams will be situate in the neighbourhood of the orifice, and their sum 
J3 may be called the virtual area of the latter. The velocity of efflux, as found from (2), is 

gr = .911Co. 

The rate of discharge is then =qp8f where q and p have the values just found, and is there- 
fore approximately independent of the external pressure Pi so long as this falls below 
-527 j>0. The ph3r8ical reason of this is (as pointed out by Reynolds) that, so long as the 
velocity at any point exceeds the velocity of sound under the conditions which obtain 
there, no change of pressure can be propagated backwards beyond this point so as to affect 
the motion higher up the stream. 

These conclusions appear to be in good agreement with experimental results. 

Under similar circumstances as to pressure, the velocities of efflux of different gases are 
(so far as y can be assumed to have the same value for each) proportional to the corre- 
sponding velocities of sound. Hence (as we shall see in Chapter x. ) the velocity of efflux will 
vary inversely, and the rate of discharge of mass will vary directly, as the square root of 
the density]:. 

Rotating Liquid, 

26. Let us next take the case of a mass of liquid rotating, under the 
action of gravity only, with constant and uniform angular velocity co about 
the axis of z, supposed drawn vertically upwards. 

* A result equivalent to this was given by Saint VenaDt and Wantzel, Jounu de VicoU 
PclffL t. xvi p. 92 (1839). 

t "On the Flow of Gases," Proc. Manch, Lit, and PhU Soc, Nov. 17, 1886; PhiL Mag, 
March 1886 [Papers, t. it p. 311]. A similar explanation was given by Hugoniot, Comptea 
Rendus, June 28, July 26, and Dec. 13, 1886. I have attempted, above, to condense the reasoning 
of these writers. 

t Cf. Graham, Phil Trans. 1846. 



26 Integration of the Eqtuitions in Special Cases [chap, n 

By hypothesis, w, v, «^ « — aiy, mXy ' 0, 

Z, Y, Z= 0, 0, -(7. 

The equation of contiimity is satisfied identically, and the dynamical equations 
obviously are 

_„^.-!|, -„v-5|, o.-i|-,. .,» 

These have the common integral 

^ = i<u« (»* + y«) -5^2-1- const (2) 

P 

The free surface, p = const., is therefore a paraboloid of revolution about the 

axis of z, having its concavity upwards, and its latus rectum = 2^/co^. 

Q. dv du rt 

Smce 5 ?r- = 2co, 

ox oy 

a velocity-potential does not exist. A motion of this kind could not therefore 
be generated in a 'perfect' fluid, i.e, in one unable to sustain tangential 
stress. 

27. Instead of supposing the angular velocity a> to be uniform, let us 
suppose it to be a function of the distance r from the axis, and let us inquire 
what form must be assigned to this function in order that a velocity-potential 
may exist for the motion. We find 

dx dy dr* 

and in order that this may vanish we must have uyr^ «= ft, ft constant. The 
velocity at any point is then = iilr, so that the equation (2) of Art. 21 
becomes 



^ = const.-J^, (1) 

if no extraneous forces act. To find the value of ^ we have, using polar 
co-ordinates, 

ar " "' rde" r ' 
whence ^ = — /xd + const. = — /itan~*- -\- const (2) 

We have here an instance of a 'cyclic' function. A function is said to be 
'single- valued' throughout any region of space when we can assign to every 
point of that region a definite value of the function in such a way that these 
values shall form a continuous system. This is not possible with the function 
(2) ; for the value of ^, if it vary continuously, changes by — 27r/Lt as the point 
to which it refers describes a complete circuit round the origin. The general 
theory of cyclic velocity-potentials will be given in the next chapter. 



26-28] 



Rotating Liquids 



27 



If gravity act, and if the axis of z be drawn vertically upwards, we must 
add to (1) the term — gz. The form of the free surface is therefore that 
generated by the revolution of the hyperbolic curve xH = const, about the 
axis of z. 

By properly fitting together the two preceding solutions we obtain the 
case of Rankine's 'combined vortex.' Thus the motion being everywhere in 
coaxial circles, let us suppose the velocity to be equal to wr from r = to 
r == a, and to cja^r for r> a. The corresponding forms of the free surface are 
then given by ' 



and 



2 = ^(r*-O«) + 0, 



these being continuous when r = a. The depth of the central depression 
below the general level of the surface is therefore wl^a'^jg. 




28. To illustrate, by way of contrast, the case of extraneous forces not 
having a potential, let us suppose that a mass of liquid filling a right circular 
cylinder moves from rest under the action of the forces 

X^Ax + By, Y = B'x^Cy, Z«0, 

the axis of z being that of the cylinder. 

If we assume u = - ^y, t; = taXy u? =0, where o> is a function of t only, these values satisfy 
the equation of continuity and the boundary conditions. The dynamical equations are 
evideiitly 



dt pox 



d» 



\dp 



(1) 



Differentiating the first of these with respect to y, and the second with respect to x and 
subtracting, we eliminate p, and find 

J=i(B'-*) (2) 



28 Integration of the Eqtuitions in Special Cases [chap, n 

The fluid therefore rotates as a whole about the axis of z with constantly accelerated 
angular velocity, except in the particular case when B=B\ To find p, we substitute the 
value of da>/dt in (1) and integrate; we thus get 

? =i<»* (x* +y*) +i (-4a?« +2/3xy +0y«) +const., 
P 
where 2p=B+B\ 

29. Afl a final example, we will take one suggested by the theory of 
'electro-magnetic rotations.' 

If an electric current be made to pass radially from an axial wire, through a conducting 
liquid, to the walls of a metallic containing cylinder, in a uniform magnetic field, the 
extraneous forces will be of the type* 

Assuming tt = - ay, v —mx, u; =0, where o) is a function of r and t only, we have 







Bq> , uX 


\dp 
P^' 




=0. 



(«) 



Eliminating p, we obtain 

The solution of this is <a=F (t)/r* +/ (r), 

where F and / denote arbitrary functions. Since » =0 when ^ =0, we have 

i^(0)/r*+/(r)=0, 
andtherefore ^^ J^ffl - J^(0) ^ X 

where X is a function of t which vanishes for ^ =0. Substituting in (1), and integrating, we 
find 

Since p is essentially a single- valued function, we must have dK/dt =/ui, or X =/i/. Hence 
the fluid rotates with an angular velocity which varies inversely as the square of the 
distance from the axis, and increases constantly with the time. 

* It C denote the total flux of electricity outwards, per unit length of the axis, and y the 
component of the magnetic force parallel to the axis, we have fi=yCl2wp. For the history of 
such experiments see Winkelmann, Handlmch d. Physik, 2nd ed. (1905), t. v. (1), p. 430. The 
above case is specially simple, in that the forces X, F, Z have a potential (0= -m tan~^ yl^)» 
though a * cyclic' one. As a rule, in electro-magnetic rotations, the mechanical forces X, Y, Z 
have not a potential at all. 



CHAPTER III 

IRROTATIONAL MOTION 

30. The present chapter is devoted mainly to an exposition of some 
general theorems relating to the kinds of motion already considered in 
Arts. 17 — 20; viz. those in which udx-^vdy + wdz is an exact differential 
throughout a finite mass of fluid. It is convenient to begin with the 
following analysis, due to Stokes*, of the motion of a fluid element in the 
most general case. 

The component velocities at the point (x, y, z) being i«, v, tr, the relative 
velocities at an infinitely near point (x + 8x, y + 8y, z + hz) are 

^ du ^ , du ^ , du ^ 
^''^di^'^-^'dy^y^dz^'^ 



dx ^ dy ^ ^ dz ' 

5 dw^ , dw ^ , dw^ 

^"di^ + d^^'J+Tz^'-^ 



(1) 



If we write 


du 


, dv 


dw 
'^dz' 




- dw dv 
^~ dy^dz' 


du dw 
^^ dz^ dx* 


, dv du 
dx dy* 




^ dw dv 
^"dy dz' 


du dw 
"^^dz dx' 


y dv du. 



equations (1) may be written 

8w = a8x + \l&y + \glz -h \ {r^z - ^y)\ 

8t;= iA8a;+ % + i/Sz + i(C8x - f8z),l (2) 

8w = \ghx + \Jhj + cSz + \{&y - lySx). J 



* *'0n the Theories of the Internal Friction of Fluids in Motion, &c." CwnA, Phil. Trans. 
t. Tiii. (1845) [Papers, t. i. p. 80]. 

t There is here a deviation from the traditional convention. It has been customary to use 
symbols such as {, 17, ^ (Helmholtz) or io\ ia\ ta'" (Stokes) to denote the component rotations 

1 ^dw dv' 
2 

of a fluid element. The fundamental kinematical theorem is however that of Art. 32 (3), and 
the definition of (, 17, ^ now adopted in the text avoids the intrusion of an unnecessary factor 
2 (or i as the case may be) in this and in a whole series of subsequent formulae relating to vortex 
motion. It also improves the electro-magnetic analogy of Art. 148. 



Xdy'dzJ' iKdz'dz)' 2\dx dy) 



30 



Irrotational Motion 



[chap, m 



Hence the motion of a small element having the point (x, y, z) for its 
centre may be conceived as made up of three parts. 

The first part, whose components are u, t?, tc;, is a motion of transUuion 
of the element as a whole. 

The second part, expressed by the first three terms on the right-hand 
sides of the equations (2), is a motion such that, if Sx, 8^, 82; be regarded as 
current co-ordinates, every point is moving in the direction of the normal to 
that quadric of the system 

a {hxY + b {hyY + c {hzf +f8y8z -\- gSzSx + h8x8y = const., ... (3) 

on which it lies. If we refer these quadrics to their principal axes, the 
corresponding parts of the velocities parallel to these axes will be 

8u' = a'Bx\ 8v' = V8y\ 8w' = c'8z\ (4) 

if a' (8x')2 + 6' (8y')* + C (Sz')* = const. 

is what (3) becomes by the transformation. The formulae (4) express that 
the length of every line in the element parallel to x' is being elongated at 
the (positive or negative) rate a\ whilst Unes parallel to y' and z' are being 
elongated in like manner at the rates b' and c' respectively. Such a motion is 
called one of vure strain a nd the principal axes of the quadrics (3) are called 
the axes of the strain. 

The last two terms on the right-hand sides of the equations (2) express a 
rotation of the element as a whole about an instantaneous axis; the com- 
ponent angular velocities of the rotation being J^, Jt;, J{*. 

The vector whose components are ^, ly, £ may conveniently be called 
the ' vorticity' of the medium at the point {x, y, z). 

This anaiysis may be iUustrated by the so-called * laminar* motion of a liquid. Thus if 

we have a, b, c, /, g, (, rj =0, h =y^ f = -/*. 

If A represent a rectangular fluid element bounded by planes parallel to the co-ordinate 
planes, then B represents the change produced in this in a short time by the strain alone, 
and C that due to the strain pl%i8 the rotation. 



rr 



y' 




* The quantities oorresponding to \^, {yi, J^in the theory of the infinitely small displacemenia 
of a continuous medium had been interpreted by Cauchy as expressing the *mean rotations* 
of an element, Exercices d* Analyse et de Physique, t. ii. (Paris, 1841), p. 302. 



30-81] Deformation of an Element 31 

It is easily seen that the above resolution of the motion is unique. If 
we assume that the motion relative to the point (a;, y, z) can be made up of a 
strain and a rotation in which the axes and coefficients of the strain and the 
axis and angular velocity of the rotation are arbitrary, then calculating the 
relative velocities 8u, 8r, hwy we get expressions similar to those on the right- 
hand sides of (2), but with arbitrary values of a, 6, c,/, g,h,(, 17, J. Equating 
coefficients of 8a;, by, 8z, however, we find that a, b, c, Sec. must have respec- 
tively the same values as before. Hence the directions of the axes of the 
strain, the rates of extension or contraction along them, and the axis and the 
amount of the vorticity, at any point of the fluid, depend only on the state 
of relative motion at that point, and not on the position of the axes of 
reference. 

When throughout a finite portion of a fluid mass we have ^, '7, C &11 ^^^o, 
the relative motion of any element of that portion consists of a pure strain 
only, and is called 'irrotational.' 

31. The value of the integral 

J {udx + vdy + wdz). 



or 



ff dx , dy ^ dz\ , 



taken along any line ABCD, is called* the 'flow' of the fluid from A to D 
along that line. We shall denote it for shortness by / (ABCD). 

If A and D coincide, so that the line forms a closed curve, or circuit, the 
value of the integral is called the 'circulation' in that circuit. We denote 
it by / (ABCA). If in either case the integration be taken in the opposite 
direction, the signs of dx/ds, dy/ds, dz/ds will be reversed, so that we have 

I (AD) ^ - I (DA), and I (ABCA) = ^ I (ACBA). 

It is also plain that 

/ (ABCD) = / (AB) + / (BG) -h / (CD). 

Again, any surface may be divided, by a double series of lines crossing 
it, into infinitely small elements. The sum of the circulations round 
the boundaries of these elements, taken all in the 
same sense, is equal to the circulation round the 
original boundary of the surface (supposed for the 
moment to consist of a single closed curve). For, 
in the sum in question, the flow along each side 
common to two elements comes in twice, once for 
each element, but with opposite signs, and there- 
fore disappears from the result. There remain then 
only the flows along those sides which are parts of 
the original boundary; whence the truth of the 
above statement. 

♦ Sir W. Thomson, "On Vortex Motion," Edin, Trans, t. xxv. (1869) [Papers, L iv. p, 13]. 




32 



Irrotational Motion 



[chap, m 



From this it follows, by considerations of continuity, that the circulation 
round the boundary of any surface-element hS, having a given position and 
aspect, is ultimately proportional to the area of the element. 

If the element be a rectangle 8^ hz having its centre at the point (x, y, z), 
then calculating the circulation round it in the direction shewn by the arrows 
in the annexed figure, we have 

y 




and therefore 







I (AB) = {v - J {dv/dz) Sz} 8y, / (BC) = {w + i {dw/dy) Sy} Sz, 
I (CD) = - {v + J (dv/dz) 8z} Sy, I (DA) = - {w? - i (dw/dy) hy} Sz, 

In this way we infer that the circulations round the boundaries of any 

infinitely small areas 88 1, SS^, 8/S3, having their planes parallel to the 

co-ordinate planes, are 

^8Si, 7y8S„ C853 (1) 

respectively. 

Again, referring to the figure and the notation of Art. 2, we have 
/ (ABC A) = / (PBCP) + / (PCAP) + / (PABP) 
= f . iA + ^ . niA + f . wA, 

whence we infer that the circulation round the boundary of any infinitely 
small area 88 is 

{l^-\-m7i + n088 (2) 

We have here an independent proof that the quantities ^, ly, C ^^7 be regarded 
as the components of a vector. 

It will be observed that some convention is implied as to the relation 
between the sense in which the circulation round the boundary of 85 is 
estimated, and the sense of the normal (2, m, n). In order to have a clear 
understanding on this point, we shall suppose in this book that the axes of 
co-ordinates form a right-handed system ; thus if the axes of x and y point E. 



31-32] Cireulation in a Finite Circuit 33 

and K. respectively, that of a will point vertically upwards*. The aeiiBe in 
which the circulation, as given by (2), is estimated is then related to the 
ditection of the normal {I, m, n) in the mannei typified by a ligbt-handed 
screw t- 

32. Expressing now that the circulation round the edge of any finite 
surface Js equal to the sum of the circulations round the boundaries of the 
infinitely small elements into which the surface mav be divided, we have, 

by (2), 

; (uit + sij + »&)-;; (if + m, + n{) (B (3) 

or, substituting the values of f , ij, i from Art. 30, 

'<"*--*+-^)=//KI-l)+-(s-s)+«(£-|)H--'*> ■ 

where the single-integral is taken along the bounding curve, and the donble- 
integral over the surface :[■ ^° these formulae the quantities I, m, n are the 
direction-cosines of the normal drawn always on one side of the surface, 
which we may term the positive aide; the direction of integration in the 
first member is then that in which a man walking on the surface, on the 
positive side of it, and close to the edge, must proceed so as to have the 
surface always on his left hand. 

The theorem (3) or (4) may evidently be extended to a surface whose 
boundary consists of two or more closed curves, provided the integration in 
the first member be taken round each of these in the 
proper direction, according to the rule just given. 
Thus, if the surface-integral in (4) extend over the 
shaded portion of the annexed figure, the directions 
in which the circulations in the several parts of the 
boundary are to be taken are shewn by the arrows, 
the positive side of the surface being that which 
faces the reader. 

The value of the surface-integral taken over a 
closed surface is zero. 

It should be noticed that (4) is a theorem of pure mathematics, and is 
true whatever functions u, v, w may be of x, y, z, provided only they be 
continuous and difierentiable at all points of the surface§. 

* Huwell. Proc. Load. MaOi. Soe. t. iiL pp. 279, 280. Thus in the fig. of p. 32 th« axis 
tit X is Bopposed drawn towuds the reader. 

t Sse HaxweU, EUetrieily and MagiKtUm, Oxford, 1873, Art 23. 

X Thia tlieorem b due to Stokes, Smith's Prize Examinatiim Paper* /or 1864. the fiiBt 
pobliahed proof Appears to have been given by Hanbel, Zur aUftm. Tkeorie der Snee^n; der 
FlStngkeiUn, Gottingen, 1861, p. 39. That given above ie due to Lord Kelvin, tc onle p. 31. 
See ako Thomson and Tait, Nafural Philo»opky, Art. 190 (», and Maxwell, EUctrieity and 
Magntlittn, Art. 24. 

J It ia not neoesBary that their di&erential <<oefficient« shoald bi 



34 Irrotaiional Motion [chap, m 

33. The rest of this chapter is devoted to a study of the kinematical 
properties of irrotational motion in general, as defined by the equations 

f,^,C = 0, (1) 

i.e. the circulation in every infinitely small circuit is assumed to be zero. 
The existence and properties of the velocity-potential in the various cases 
that may arise will appear as consequences of this definition. 

The phjrsical importance of the subject rests on the fact that if the 
motion of any portion of a fluid mass be irrotational at any one instant it 
will under certain very general conditions continue to be irrotational. 
Practically, as will be seen, this has already been established by Lagrange's 
theorem, proved in Art. 17, but the importance of the matter warrants a 
repetition of the investigation, in terms of the Eulerian notation, in the form 
given by Lord Kelvin*. 

Consider first any terminated line AB drawn in the fluid, and suppose 
every point of this line to move always with the velocity of the fluid at that 
point. Let us calculate the rate at which the flow along this line, from A to 
B, is increasing. If 8x, Sy, 8z be the projections on the co-ordinate axes of 
an element of the line, we have 

-(uSx)=-^8x + u-j^. 

Now DSx/Dt, the rate at which Sx is increasing in consequence of the motion 
of the fluid, is equal to the difference of the velocities parallel to x at the 
two ends of the element, i,e, to 8u ; and the value of Du/Dt is given by Art. 5. 
Hence, and by similar considerations, we find, if p be a function of p only, 
and if the extraneous forces X, Y, Z have a potential H, 

=r- (u8x + vSv + w8z) = — ^ — 8n -i- u8u -h v8v + t(f8w. 
Dt "^ p 

Integrating along the line, from ^4 to 5, we get 



^W^ ^|^(w(fa + vdy+M;(iz)=|^-|^-n4-i?* \ 



(2) 



or, the rate at which the flow from il to jB is increasing is equal to the excess 
of the value which — Sdp/p — fl + i?* has at B over that which it has at A, 
This theorem comprehends the whole of the dynamics of a perfect fluid. For 
instance, equations (2) of Art. 15 may be derived from it by taking as the 
line AB the infinitely short line whose projections were originally 8a, 86, 8c, 
and equating separately to zero the coeflScients of these infinitmmals. 

If ft be single-valued, the expression within brackets on the right-hand 
side of (2) is a single-valued function of x, y, z. Hence if the integration on 

* Ic. ante p. 31. 



33-35] Vdoeity-Fotential 35 

the left-hand be taken round a closed curve, bo that B coincides with A, 
we have 



D f 

yjT I {udx + vdy + wdz) = 0, (3) 



Dt 

or, the circulation in any circuit moving with the fluid does not alter with 
the time. 

It follows that if the motion of any portion of a fluid mass be initially 
irrotational it will always retain this property ; for otherwise the circulation 
in every infinitely small circuit would not continue to be zero, as it is initially 
by virtue of Art. 32 (3). 

34. Considering now any region occupied by irrotationally-moving fluid, 
we see from Art. 32 (3) that the circulation is zero in every circuit which 
can be filled up by a continuous surface lying wholly in the region, or which 
in other words is capable of being contracted to a point without passing out 
of the region. Such a circuit is said to be 'reducible.' 

Again, let us consider two paths ACB, ADB, connecting two points A, B 
of the region, and such that either may by continuous variation be made to 
coincide with the other, without ever passing out of the region. Such paths 
are called 'mutually reconcileable.' Since the circuit ACBDA is reducible, 
we have / {ACBDA) = 0, or since / (BDA) = - / (ADB), 

I (ACB) ^ I (ADB); 

i.e. the flow is the same along any two reconcileable paths. 

A region such that aU paths joining any two points of it are mutually 
reconcileable is said to be 'simply-connected.' Such a region is that enclosed 
within a sphere, or that included between two concentric spheres. In what 
follows, as far as Art. 46, we contemplate only simply-connected regions. 

35. The irrotational motion of a fluid within a simply-connected region 
is characterized by the existence of a single-valued velocity-potential. Let 
us denote by — ^ the flow to a variable point P from some fixed point A, viz. 

^ = — I {udx -{- vdy -^ wdz) (1) 

The value of <f> has been shewn to be independent of the path along which 
the integration is effected, provided it lie wholly within the region. Hence 
^ is a single- valued function of the position of P ; let us suppose it expressed 
in terms of the co-ordinates (x, y, z) of that point. By displacing P through 
an infinitely short space parallel to each of the axes of co-ordinates in 
succession, we find 

»-£■ "--| "-I '^' ■ 

i,e. ^ is a velocity-potential, according to the definition of Art. 17. 

3—2 



36 Irrotational Motion [chap, in 

The substitution of any other point B for A, as the lower limit of the 
integral in (1), simply adds an arbitrary constant to the value of <f>, viz. the 
flow from A to B. The original definition of <f> in Art. 17, and the physical 
interpretation in Art. 18, alike leave the function indeterminate to the extent 
of an additive constant. 

As we follow the course of any line of motion the value of (f> continually 
decreases; hence in a simply-connected region the lines of motion cannot 
form closed curves. 

36. The function </> with which we have here to do is, together with its 
first differential coefficients, by the nature of the case, finite, continuous, and 
single-valued at all points of the region considered. In the case of incom- 
pressible fluids, which we now proceed to consider more particidarly, <f} must 
also satisfy the equation of continuity, (5) of Art. 20, or as we shall in future 

write it, for shortness, 

VV = 0, (1) 

at every point of the region. Hence ^ is now subject to mathematical 
conditions identical with those satisfied by the potential of masses attracting 
or repelling according to the law of the inverse square of the distance, at all 
points external to such masses; so that many of the results proved in the 
theories of Attractions, Electrostatics, Magnetism, and the Steady Flow of 
Heat, have also a hydrodynamical application. We proceed to develop those 
which are most important from this point of view. 

In any case of motion of an incompressible fluid the surface-integral of 
the normal velocity taken over any surface, open or closed, is conveniently 
called the *flux' across that surface. It is of course equal to the volume of 
fluid crossing the surface per unit time. 

When the motion is irrotational, the flux is given by 

d<f> 



-I! 



dn^' 



where SS is an element of the surface, and Sn an element of the normal to it^ 
drawn in the proper direction. In any region occupied whoUy by liquid, the 
total flux across the boundary is zero, i.e. 



II 



&■«»-». (2) 



the element 8w of the normal being drawn always on one side (say inwards), 
and the integration extending over the whole boundary. This may be 
regarded as a generaUzed form of the equation of continuity (1). 

The lines of motion drawn through the various points of an infinitesimal 
circuit define a tube, which may be called a tube of flow. The product of 
the velocity (q) into the cross-section (a, say) is the same at all points of such 
a tube. 



35-38] Ttibes of Flow 37 

We may, if we choose, regard the whole space occupied by the fluid as 
made up of tubes of flow, and suppose the size of the tubes so adjusted that 
the product qa is the same for eiach. The flux across any surface is then 
proportional to the number of tubes which cross it. If the surface be closed, 
the equation (2) expresses the fact that as many tubes cross the surface 
inwards as outwards. Hence a line of motion cannot begin 'or end at a point 
of the fluid, 

37. The function <f} cannot be a maximum or a minimum at a point in the 
interior of the fluid ; for, if it were, we should have d{f>/dn everywhere positive, 
or everywhere negative, over a small closed surface surrounding the point in 
question. Either of these suppositions is inconsistent with (2). 

Further, the square of the velocity cannot be a mcucimum at a point 
in the interior of the fluid. For let the axis of x be taken parallel to the 
direction of the velocity at any point P. The equation (1), and therefore 
abo the equation (2), is satisfied if we write d<f)/dx for </>, The above 
argument then shews that d<f>/dx cannot be a maximum or a minimum at P. 
Hence there must be points in the immediate neighbourhood of P at which 
(3^/3aj)* and therefore a fortiori 



(S)'+(l)'+^^' 



,dyj \dzj 
is greater than the square of the velocity at P*. 

On the other hand, the square of the velocity may be a minimum at 
some point of the fluid. The simplest case is that of a zero velocity; see, 
for example, the figure of Art. 69, below. 

38. Let us apply (2) to the boundary of a finite spherical portion of the 
liquid. If r denote the distance of any point from the centre of the sphere, 
hm the elementary solid angle subtended at the centre by an element 8S of 
the surface, we have 

d(f}/dn = — d<f>/dr, 

and 85 = r^to. Omitting the factor r*, (2) becomes 



|<i»-0, 



// 

or ^jj<f>dm = (3) 

Since l/iir . ff<f}dm or l/47rr* - Si<f>dS measures the mean value of </> over 
the surface of the sphere, (3) shews that this mean value is independent of 

* This theorem was enunciated, in another connection, by Lord Kelvin, Phil. Mag, Oct. 
1S60 [Reprint of Papers an Electrogtalies, dtc, London, 1872, Art, 606]. The above demon- 
stration is due to Kirchho£f, Vorlesungen Hher mathemaiiwhe Phyeik, Meehanik^ Leipzig, 1876, 
p. 186. For another proof see Art. 44 below. 



38 Irrotdtional Motion [chap, in 

the radius. It is therefore the same for any sphere, concentric with the 
former one, which can be made to coincide with it by gradual variation of the 
radius, without ever passing out of the region occupied by the irrotationally 
moving liqui<i. We may therefore suppose the sphere contracted to a point, 
and so obtain a simple proof of the theorem, first given by Gauss in his 
memoir* on the theory of Attractions, that the mean value of ^ over any 
spherical surface throughout the interior of which (1) is satisfied, is equal to 
its value at the centre. 

The theorem, proved in Art. 37, that <f> cannot be a maximum or a 
minimum at a point in the interior of the fluid, is an obvious consequence of 
the above. 

The above proof appears to be due, in principle, to Frost f. Another 
demonstration, somewhat different in form, has been given by Lord Rayleighij:. 
The equation (1), being linear, will be satisfied by the arithmetic mean of any 

number of separate solutions ^x, ^2' ^s* ^^ ^ suppose an infinite 

number of systems of rectangular axes to be arranged uniformly about any 
point P as origin, and let <f>i,<f>29<f>3f • • - ^^ the velocity-potentials of motions 
which are the same with respect to these several systems as the original 
motion <f> is with respect to the system a;, y, z. In this case the arithmetic 
mean (^, say) of the functions <f>i, ^2> ^3' * • * ^^^ ^^ ^ function of r, the 
distance from P, only. Expressing that in the motion (if any) represented 
by 4>i the flux across any spherical surface which can be contracted to a point, 
without passing out of the region occupied by the fluid, would be zero, we have 



^.'*.o. 



or ^ = const. 



39. Again, let us suppose that the region occupied by the irrotationally 
moving fluid is *periphractic,'§ i.e. that it is limited internally by one or more 
closed surfaces, and let us apply (2) to the space included between one (or 
more) of these internal boundaries, and a spherical surface completely 
enclosing it (or them) and Ijring wholly in the fluid. If M denote the total 
flux into this region, across the internal boundary, we find, with the same 
notation as before. 



// 



^iS.-U. 



* "Allgemeine Lehrsatze, u.8.w.," ResuUaie aua den Bedbachtungen des magnetischen 
Vereins, 1839 [Werke, Qottingen, 1870-80, t. v. p. 199]. 

t Quarterly Journal of MaiJiematieaf t. zii (1873). 

t Messenger of Maihematics, t. vii p. 69 (1878) [ScierUific Papers, Cambridge, 1899. . ., t. L 
p. 347]. 

§ See Maxwell, Electricity and Magnetism, Arte. 18, 22. A region is said to be 'aperiphractic* 
when every closed suiface drawn in it can be contracted to a point without passing out of the 
region. 



38-40] Mean Value over a Spherical Surface 39 

the suifaoe-integrsl extending over the sphere only. This may be written 

M 



^I.ff4,dm = - 



4ffr»' 



whence ^J|^eiS = ^//^de = ^+ C (4) 

That is, the mean value of <f} over any spherical surface drawn under the 
above-mentioned conditions is equal to M/4^rrr + C, where r is the radius, M 
an absolute constant, and C a quantity which is independent, of the radius 
but may vary with the position of the centre*. 

If however the original region throughout which' the irrotational motion 
holds be unlimited externally, and if the first derivative (and therefore all the 
higher derivatives) of <f> vanish at infinity, then C is the same for all spherical 
surfaces enclosing the whole of the internal boundaries. For if such a sphere 
be displaced parallel to xf, without alteration of size, the rate at which C 
varies in consequence of this displacement is, by (4), equal to the mean value 
of d<f)/dx over the surface. Since d<f>/dx vanishes at infinity, we can by taking 
the sphere large enough make the latter mean value as small as we please. 
Hence C is not altered by a displacement of the centre of the sphere parallel 
to X. In the same way we see that C is not altered by a displacement parallel 
to y or 2; i.e. it is absolutely constant. 

If the internal boundaries of the region considered be such that the total 
flux across them is zero, e.g. if they be the surfaces of solids, or of portions of 
incompressible fluid whose motion is rotational, we have if = 0, so that the 
mean value of <f> over any spherical surface enclosing them all is the same. 

40. (a) If <f> be constant over the boundary of any simply-connected 
region occupied by liquid moving irrotationally, it has the same constant 
value throughout the interior of that region. For if not constant it 
would necessarily have a maximum or a minimum value at some point 
of the region. 

Otherwise : we have seen in Arts. 35, 36 that the lines of motion cannot 
begin or end at any point of the region, and that they cannot form closed 
curves Ijdng wholly within it. They must therefore traverse the region, 
beginning and ending on its boundary. In our case however this is impossible, 
for such a line always proceeds from places where <f> is greater to places where 
it is less. Hence there can be no motion, i.e. 

d<f> o<p d(f> -. 

dx' 8^' di^ ' 

and therefore <f> is constant and equal to its value at the boundary. 

^ It is imdentoody of course, that the spherical surfaces to which this statement applies are 
reconoileable with one another, in a sense analogous to that of Art. 34. 
t Kirchhofif, Meehanik, p. 191. 



40 IrrotcUional Motion [chap, in 

(P) Again, if d^jdn be zero at every point of the boundary of such a 
region as is above described, <f> will be constant throughout the interior. For 
the condition d^jdn = expresses that no lines of motion enter or leave the 
region, but that they are all contained within it. This is however, as we 
have seen, inconsistent with the other conditions which the lines must 
conform to. Hence, as before, there can be no motion, and (f> is constant. 

This theorem may be otherwise stated as follows : no continuous irrota- 
tional motion of a liquid can take place in a simply-connected region bounded 
entirely by fixed rigid walls. 

(y) Again, let the boundary of the region considered consist partly of 
surfaces 8 over which <f> has a given constant value, and partly of other 
surfaces 2 over which d<f>/dn = 0. By the previous argument, no lines of 
motion can pass from one point to another of S, and none can cross S. Hence 
no such lines exist ; <f> is therefore constant as before, and equal to its value 
at/S. 

It follows from these theorems that the irrotational motion of a liquid in 
a simply-connected region is determined when either the value of <f>, or the 
value of the inward normal velocity — d<f>/dn, is prescribed at aU points of the 
boundary, or (again) when the value of <f> is given over part of the boundary, 
and the value of — d<^/dn over the remainder. For ii<f}j^, <f>2 be the velocity- 
potentials of two motions each of which satisfies the prescribed boundary- 
conditions, in any one of these cases, the functional — <f>2 satisfies the condition 
(a) or (P) or (y) of the present Article, and must therefore be constant 
throughout the region. 

41. A glass of cases of great importance, but not strictly included in the 
scope of the foregoing theorems, occurs when the region occupied by the 
irrotationally moving liquid extends to infinity, but is bounded internally by 
one or more closed surfaces. We assume, for the present, that this region is 
simply-connected, and that <f> is therefore single- valued. 

If if) be constant over the internal boundary of the region, and tend every- 
where to the same constant value at an infinite distance from the internal 
boundary, it is constant throughout the region. For otherwise <f> would be a 
maximum or a minimum at some point within the region. 

We. infer, exactly as in Art. 40, that if <f> be given arbitrarily over the 
internal boundary, and have a given constant value at infinity, its value is 
everywhere determinate. 

Of more importance in our present subject is the theorem that, if the 
normal velocity be zero at every point of the internal boundary, and if the 
fluid be at rest at infinity, then <f} is everywhere constant. We cannot how- 
ever infer this at once from the proof of the corresponding theorem in Art. 40. 
It is true that we may suppose the region limited externally by an infinitely 



40-41] Conditions of Determinateness 41 

large surface at every point of which d<f}/dn is infinitely small; but it is 
conceivable that the integral i[d<f>jdn . dS, taken over a portion of this surface, 
might still be finite, in which case the investigation referred to would fail. 
We proceed therefore as follows. 

Since the velocity tends to the limit zero at an infinite distance from the 
internal boundary (£», say), it must be possible to draw a closed surface S 
completely enclosing S, beyond which the velocity is everywhere less than a 
certain value e, which value may, by making IE large enough, be made as 
9mall as we please. Nov in any direction from S let us take a point P at 
such a distance beyond % that the soUd angle which S subtends at it is 
infinitely small ; and with P as centre let us describe two spheres, one just 
excluding, the other just including S, We shall prove that the mean value 
of <f> over each of these spheres is, within an infinitely small amount, the 
same. For if Q, Q' be points of these spheres on a common radius PQQ\ then 
if Q, Q' fall within 2 the corresponding values of </> may differ by a finite 
amount ; but since the portion of either spherical surface which falls within S 
is an infinitely small fraction of the whole, no finite difference in the mean 
values can arise from this cause. On the other hand, when Q, Q' fall without 
2, the corresponding values of <f> cannot differ by so much as € . QQ\ for € is 
by definition a superior limit to the rate of variation of ^. Hence, the mean 
values of <f> over the two spherical surfaces must differ by less than € . QQ\ 
Since QQ' is finite, whilst € may by taking 2 large enough be made as small 
as we please, the difference of the mean values may, by taking P sufficiently 
distant, be made infinitely small. 

Now we have seen in Arts. 38, 39 that the mean value of <f> over the 
inner sphere is equal to its value at P, and that the mean value over the 
outer sphere is (since ilf = 0) equal to a constant quantity C. Hence, 
ultimately, the value of i at infinity tends everywhere to the constant 
value C. 

The same result holds even if the normal velocity be not zero over the 
internal boundary ; for in the theorem of Art. 39 ilf is divided by r, which is 
in our case infinite. 

It follows that if d<f>ldn = at all points of the internal boundary, and if 
%he fluid be at rest at infiinity, it must be everywhere at rest. For no lines 
of motion can begin or end on the internal boundary. Hence such lines, if 
they existed, must come from an infinite distance, traverse the region occupied 
by the fluid, and pass off ag^in to infinity; i,e, they must form infinitely long 
courses between places where <f> has, within an infinitely small amount, the 
same value C, which is impossible. 

The theorem that, if the fluid be at rest at infinity, the motion is deter- 
minate when the value of — 3^/9n is given over the internal boundary, follows 
by the same argument as in Art. 40. 



42 Irrotational Motion [chap, m 

Green's Theorem. 

42. In treatises on Electrostatics, &c., many important properties of 
the potential are usually proved by means of a certain theorem due to Green. 
Of these the most important from our present point' of view have already 
been given ; but as the theorem in question leads, amongst other things, to a 
useful expression for the kinetic energy in any case of irrotational motion, 
some account of it will properly find a place here. 

Let U, F, W be any three functions which are finite, single-valued and 
differentiable at all points of a connected region completely bounded by one 
or more closed surfaces S ; let 8S be an element of any one of these surfaces, 
and I, m, n the direction-cosines of the normals to it drawn inwards. We 
shall prove in the first place that 

ffilU + mV + nW)dS^ "///(^ "*" 1^ + ^)dxdydz, ... .(1) 

where the triple-integral is taken throughout the region, and the double- 
integral over its boundary. 

If we conceive a series of surfaces drawn so as to divide the re^on into 
any number of separate parts, the integral 

saw -{-mV + nW)dS, ,..,(2) 

taken over the original boundary, is equal to the sum of the similar integrals 
each taken over the whole boundary of one of these parts. For, for every 
element Sa of a dividing surface, we have, in the integrals corresponding to 
the parts lying on the two sides of this surface, elements {lU + mV + vbW) 8a, 
and {VU + m^V + n'W) 8<7, respectively. But the normals to which ?, m, n 
and V, m\ n' refer being drawn inwards in each case, we have Z' = — Z, w' = — w, 
7(' = — n ; so that, in forming the sum of the integrals spoken of, the elements 
due to the dividing surfaces disappear, and we have left only those due to 
the original boundary of the region. 

Now let us suppose the dividing surfaces to consist of three systems of 
planes, drawn at infinitesimal intervals, parallel to yZy zx, xy, respectively. If 
X, y, z be the co-ordinates of the centre of one of the rectangular spaces thus 
formed, and 8x, Sy, Sz the lengths of its edges, the part of the integral (2) due 
to the yz-iskoe nearest the origin is 

and that due to the opposite face is 

The sum of these is —dU/dx ,8xSy8z. Calculating in the same way the 



42-43] Greeii's Theorem 43 

parts of the integral due to the remaining pairs of faces, we get for the final 
result 

Hence (1) simply expresses the fact that the surface-integral (2), taken over 
the boundary of the region, is equal to the sum of the similar integrals taken 
over the boundaries of the elementary spaces of which we have supposed it 
built up. 

It is evident from (1), or it may be proved directly by transformation of 
co-ordinates, that if ?7, F, TT be regarded as components of a vector, the 
expression 

^jo_ ^y dw_ 

dx dy dz 

is a 'scalar' quantity, i.e. its value is unaffected by any such transformation. 
It is now usually called the * divergence' of the vector-field at the point 
(Xy y, z). 

The interpretation of (1), when (Z7, F, W) is the velocity of a continuous 
substance, is obvious. In the particular case of irrotational motion we obtain 

jj^dS=-IJfvhl,dxdydz (3) 

where Sn denotes an element of the inwardly-directed normal to the surface S. 

Again, if we put U, Vj W ^ pUy pv, pw, respectively, we reproduce in 
substance the second investigation of Art. 7. 

Another useful result is obtained by putting ?7, F, TF = u<f>, t^, w<f}, 
respectively, where w, v, w satisfy the relation 

3i* 9t; 3tt? _ ^ 

dx dy dz 
throughout the region, and make 

lu + mv + nw = 
over the boundary. We find 

///(»! +"i +"!)*'**-»• w 

The function <f> is here merely restricted to be finite, single- valued, and con- 
tinuous, and to have its first differential coefficients finite, throughout the 
region. 

43. Now let <f>, <f>^ be any two functions which, together with their first 
and second derivatives, are finite and single-valued throughout the region 
considered; and let us put 

respectively, so that lU + mV + nW = ^ -^^ 



44 Irrotational Motion [chap, m 

$ub6titutmg in (1) we find 

- lll^^^' dxdydz (5) 

By interchanging ^ and <f> we obtain 

-- lll<f>'Vhf>dxdydz (6) 

Equations (5) and (6) together constitute Green's theorem*. 

44. If <f>y if}' be the velocity-potentials of two distinct modes of irrotational 
motion of a Uquid, so that 

VV = 0, V^' = 0, (1) 

we ob tain jh^^'^^j^'^^ ^^ ^ 

If we recall the physical interpretation of the velocity-potential, given in 
Art. 18, then, regarding the motion as generated in each case impulsively 
from rest, we recognize this equation as a particular case of the dynamical 
theorem that 

where p^, qr and p/ , q/ are generalized components of impulse and velocity, 
in any two possible motions of a system f. 

Again, in Art. 43 (6) let </>' = ^, and let <f} be the velocity-potential of a 
liquid. We obtain 

To interpret this we multiply both sides by J/o. Then on the right-hund 
side — d<f}/dn denotes the normal velocity of the fluid inwards, whilst p(f} is, by 
Art. 18, the impulsive pressure necessary to generate the motion. It is a 
proposition in Dynamics j: that the work done by an impulse is measured by 
the product of the impulse into half the sum of the initial and final velocities, 
resolved ip. the direction of the impulse, of the point to which it is applied. 
Hence the right-hand side of (3), when modified as described, expresses the 
work done by the system of impulsive pressures which, applied to the surface 
S, would generate the actual motion; whilst the left-hand side gives the 
kinetic energy of this motion. The formula asserts that these two quantities 

* 0. Green* EsMy on Electricity and Magnetism, Nottingham, 1828, Art. 3 [McUhematical 
Papers (ed. Ferreta), Cambridge, 1871, p. 23]. 

t Thomson and Tait, Natural PhUoeophy, Art. 313, equation (11). 
t Ibid, Art. 308. 



43-45] Kinetic Energy 45 

are equal. Hence if T denote the total kinetic energy of the liquid, we have 
the very important result 

2^=-^/Mt'^ ••••• w 

If in (3), in place of <^, we write dfftjdx, which will of course satisfy V*9^/3x=0, and 
apply the resulting theorem to the region included within a spherical surface of radius r 
having any point (x, y, z) as centre, then with the same notation as in Art. 39, we have 

Hence, writing q* = u^ + v^+w^. 

Since this latter expression is essentially positive, the mecin value of g', taken over a 
sphere having any given point as centre, increases with the radius of the sphere. Hence 
9* cannot he a maximum at any poiAt of the fluid, as was proved otherwise in Art. 37. 

Moreover, recalling the formula for the pressure in any case of irrotational motion of a 
liquid, viz. 

?=|_O-l^.i-(0 (6) 

we infer that, provided the potential Q of the external forces satisfy the condition 

V«Q=0 (7) 

the mean value of p over a sphere described with any point in the interior of the fluid as 
centre will diminish as the radius increases. The place of least pressure will therefore be 
somewhere on the boundary of the fluid. This has a bearing on the point discussed in 
Art. 23. 

45. In this connection we may note a remarkable theorem discovered by 
Lord Kelvin*, and afterwards generalized by him into an universal property 
of dynamical systems started impulsively from rest under prescribed velocity- 
conditions f. 

The irrotational motion of a liquid occupying a simply-connected region 
has less kinetic energy than any other motion consistent with the same 
normal motion of the boundary. 

Let T be the kinetic energy of the irrotational motion to which the 
velocity-potential <f> refers, and Tj that of another motion given by 

^"~ax ^^' ^^"■^■^^«' ^=-^ + ^0' (®^ 

♦ (W. Thomaon) "On the Vis- Viva of a liquid in Motion," Camb. and Dub. Math, Joum. 
1849 [Papers, t. i. p. 107]. 

t Thomson and Tait, Art. 312. 



46 Irrotational Motion [chap, in 

where, in virtue of the equation of continuity, and the prescribed boundary- 
' condition, we must have 

3a? dy dz 
throughout the region, and Iuq + mvq + nw^ = 
over the boundary. Further let us write 

^0 = i/> Hi (V + V + W7o«) dxdydz. (9) 

We find Ti = T + To - /> ///(^o ^^-^ ''o^ + ^o^^) dxdydz. 

Since the last integral vanishes, by Art. 42 (4), we have 

T, = T+To, (10) 

which proves the theorem*. 

46. We shall require to know, hereafter, the form assumed by the ex- 
pression (4) for the kinetic energy when the fluid extends to infinity and is 
at rest there, being limited internally by one or more closed surfaces S. Let 
us suppose a large closed surface 2 described so as to enclose the whole of S. 
The energy of the fluid included between S and 2 is 

-},//*g^_j.//*i^, (11) 

where the integration in the first term extends over £•, that in the second 
over 2. Since we have, by the equation of continuity, 

//l^ + Z/S^-O' 

the expression (11) may be written 

-yjj(^-c)^ds-yjj{<f>-c)^^di, (12) 

where C may be any constant, but is here supposed to be the constant value 
to which <f> was shewn in Art. 39 to tend at an infinite distance from S, 
Now the whole region occupied by the fluid may be supposed made up of 
tubes of flow, each of which must pass either from one point of the internal 
boundary to another, or from that boundary to infinity. Hence the value of 
the integral 



// 



K^- 



taken over any surface, open or closed, finite or infinite, drawn within the 
region, must be finite. Hence ultimately, when 2 is taken infinitely large 
and infinitely distant all round from S, the second term of (12) vanishes, and 
we have 



2T=^-pjj{<f>-C)^^dS, (13) 



where the integration extends over the internal boundary only. 

* Some eztensionB of this result are discussed by Leathern, Cambridge Tracts, No. 1« 2nd ed. 
(1913). They supply further interesting illustrations of Kelvin's general dynamical principle. 



45-47] Cyclic Regions 47 

If the total flux across the internal boundary be zero, we have 

so that (13) may be written 2T = - p (U ^^ dS, (14) 

simply. 

On MuUiply-connect^ Regions. 

47. Before discussing the properties of irrotational motion in multiply- 
connected regions we must examine more in detail the nature and classification 
of such regions. In the following synopsis of this branch of the geometry of 
position we recapitulate for the sake of completeness one or two definitions 
already given. 

We consider any connected region of space, enclosed by boundaries. 
A region is 'connected' when it is possible to pass from any one point of 
it to any other by an infinity of paths, each of which lies wholly in the 
region. 

Any two such paths, or any two circuits, which can by continuous 
variation be made to coincide without ever passing out of the region, are said 
to be 'mutually reconcileable.' Any circuit which can be contracted to 
a point without passing out of the region is said to be 'reducible.' Two 
reconcileable paths, combined, form a reducible circuit. If two paths or two 
circuits be reconcileable, it must be possible to connect them by a continuous 
surface, which lies wholly within the region, and of which they form the 
complete boundary ; and conversely. 

It is further convenient to distinguish between 'simple' and 'multjiple' 
irreducible circuits. A 'multiple' circuit is one which can by continuous 
variation be made to appear, in whole or in part, as the repetition of another 
circuit a certain number of times. A 'simple' circuit is one with which this 
is not possible. 

A 'barrier,' or 'diaphragm,' is a surface drawn across the region, and 
limited by the line or lines in which it meets the boundary. Hence a barrier 
is necessarily a connected surface, and cannot consist of two or more detached 
portions. 

A 'simply-connected' region is one such that all paths joining any two 
points of it are reconcileable, or such that all circuits drawn within it are 
reducible. 

A 'doubly-connected' region is one such that two irreconcileable paths, 
and no more, can be drawn between any two points Ay B of it ; viz. any other 
path joining AB is reconcileable with one of these, or with a combination of 
the two taken each a certain number of times. In other words, the region is 



48 Irrotational Motion [chap, hi 

such that one (simple) irreducible circuit can be drawn in it, whilst all other 
circuits are either reconcileable with this (repeated, if necessary), or are 
reducible. As an example of a doubly-connected region we may take that 
enclosed by the surface of an anchor-ring, or that external to such a ring and 
extending to infinity. 

Generally, a region such that n irreconcileable paths, and no more, can be 
drawn between any two points of it, or such that n — \ (simple) irreducible 
and irreconcileable circuits, and no more, can be drawn in it, is said to be 
' w-ply-connected.' 

The shaded portion of the figure on p. 33 is a triply-connected space of 
two dimensions. 

It may be shewn that the above definition of an n-ply-connected space 
is self-consistent. In such simple cases as n = 2, n = 3, this is sufficiently 
evident without demonstration. 

48. Let us suppose, now, that we have an n-ply-connected region, with 
n — \ simple independent irreducible circuits drawn in it. It is possible to 
draw a barrier meeting any one of these circuits in one point only, and not 
meeting any of the w — 2 remaining circuits. A barrier drawn in this manner 
does not destroy the continuity of the region, for the interrupted circuit 
remains as a path leading round from one side to the other. The order of 
connection of the region is however diminished by unity; for every circuit 
drawn in the modified region must be reconcileable with one or more of the 
w — 2 circuits not met by the barrier. 

A second barrier, drawn in the same manner, will reduce the order of 
connection again by one, and so on ; so that by drawing w — 1 barriers we can 
reduce the region to a simply-connected one. 

A simply-connected region is divided by a barrier into two separate 
parts; for otherwise it would be possible to pass from a point on one side 
the barrier to an adjacent point on the other side by a path Ipng wholly 
within the region, which path would in the original region form an irreducible 
circuit. 

Hence in an n-ply-connected region it is possible to draw n — 1 barriers, 
and no more, without destropng the continuity of the region. This property 
is sometimes adopted as the definition of an n-ply-connected space. 

Irrotational Motion tn MuUiply-conneded Spaces. 

49. The circulation is the same in any two reconcileable circuits ABCA, 
A'B'C'A' drawn in a region occupied by fluid moving irrotationally. For the 
two circuits may be connected by a continuous surface lying wholly within 
the region; and if we apply the theorem of Art. 82 to this surface, we 



47-50] Cydic Vdoeity-Potentials 49 

have, remembering the rule as to the direction of integration round the 

boundary, 

I (ABC A) + / (A'CWA') = 0, 

or I (ABC A) = I (A'WC'A'). 

If a circuit ABC A be reconcileable with two or more circuits A'B'C'A\ 
A"B"C"A'\ &c., combined, we can connect all these circuits by a continuous 
surface which lies wholly within the region, and of which they form the com- 
plete boundary. Hence 

/ (ABC A) + I (A'C'B'A') + I (A"C''B''A") + &c. = 0, 
or I (ABC A) = I(A'B'C'A') + / (A"B'VA'') + &c. ; 

i.e. the circulation in any circuit is equal to the sum of the circulations in the 
several members of any set of circuits with which it is reconcileable. 

Let the order of connection of the region be n + 1, so that n independent 
simple irreducible circuits ^i , ag , ... a„ can be drawn in it ; and let the circu- 
lations in these be ic ^ , ktj , ... #f „ , respectively. The sign of any k will of course 
depend on the direction of integration round the corresponding circuit; let 
the direction in which k is estimated be called the positive direction in the 
circuit. The value of the circulation in any other circuit can now be found 
at once. For the given circuit is necessarily reconcileable with some com- 
bination of the circuits ai^a^, ... a„; say with a^ taken j>^ times, a, taken 
Pj times and so on, where of course any p is negative when the corre- 
sponding circuit is taken in the negative direction. The required circulation 
then is 

Pl*^! + P^t + . . . + P«^n (1) 

Since any two paths joining two points A, B of the region together form 
a circuit, it follows that the values of the flow in the two paths differ by 
a quantity of the form (1), where, of course, in particular cases some or all of 
the j7's may be zero. 

50. Let us denote by — ^ the flow to a variable point P from a fixed 
point -4, viz. 

<^ = — I (udx -I- vdy + wdz) (2) 

So long as the path of integration from ^ to P is not specified, <^ is indeter- 
minate to the extent of a quantity of the form (1). 

If however n barriers be drawn in the manner explained in Art. 48, so as 
to reduce the region to a simply-connected one, and if the path of integration 
in (2) be restricted to lie within the region as thus modified (Le. it is not to 
cross any of the barriers), then <^ becomes a single-valued function, as in 
Art. 35. It is continuous throughout the modified region, but its values at 
two adjacent points on opposite sides of a barrier differ by ± k. To derive 
the value of <f> when the integration is taken along any path in the unmodified 
region we must subtract the quantity (1), where any p denotes the number of 

L. H. 4 



60 Irrotational Motion [chap, hi 

times this path crosses the corresponding barrier. A crossing in the positive 
direction of the circuits interrupted by the barrier is here counted as positive, 
a crossing in the opposite direction as negative. 

By displacing P through an infinitely short space parallel to each 
co-ordinate axis in succession, we find 

3<i dfj> dd) 

SO that <f> satisfies the definition of a velocity-potential (Art. 17). It is now 
however a many- valued or cycUc function ; i.e. it is not possible to assign to 
every point of the original region a unique and definite value of <f>, such 
values forming a continuous system. On the contrary, whenever P describes 
an irreducible circuit, <f> will not, in general, return to its original value, but 
will differ from it by a quantity of the form (1). The quantities ici, icj* • • • '^m 
which specify the amounts by which <f> decreases as P describes the several 
independent circuits of the region, may be called the *cycUc constants' of <f>. 

It is an immediate consequence of the * circulation-theorem' of Art. 33 
that under the conditions there presupposed the cyclic constants do not alter 
with the time. The necessity for these conditions is exemplified in the 
problem of Art. 29, where the potential of the extraneous forces is itself 
a cycUc function. 

The foregoing theory may be illustrated by the case of Art. 27 (2), where the region (as 
limited by the exolusion of the origin, where the formula would give an infinite velocity) 
is doubly-connected ; since we can connect any two points ^, B of it by two irreconcileable 
X>aths passing on opposite sides of the axis of z, e.g. 
ACBt ADB in the figure. The portion of the plane 
zx for which x is positive may be taken as a barrier, 
and the region is thus made simply-connected. The 
circulation in any circuit meeting this barrier once 
only, e,g. in ACBDA, is 

fi/r .rdS, or 2iru,. 



That in any circuit not meeting the barrier is zero. In the modified region (f) may be put 
equal to a single- valued function, viz. -ftB, but its value on the positive side of the barrier 
is zero, that at an adjacent point on the negative side is - 2rrfx. 

More complex illustrations of irrotational motion in multiply-connected spaces of two 
dimensions will present themselves in the next chapter. 

51. Before proceeding further we may briefly indicate a somewhat 
different method of presenting the above theory. 

Starting from the existence of a velocity-potential as the characteristic 
of the class of motions which we propose to study, and adopting the second 
definition of an w -f 1 -ply-connected region, indicated in Art. 48, we remark 
that in a simply-connected region every equipotential surface must either be 
a closed surface, or else form a barrier dividing the region into two separate 



J i 




60-52 J Multiple Connectivity 51 

parts. Hence, supposing the whole system of such surfaces drawn, we see 
that if a closed curve cross any given equipotential surface once it must cross 
it again, and in the opposite direction. Hence, corresponding to any element 
of the curve, included between two consecutive equipotential surfaces, we 
have a second element such that the flow along it, being equal to the 
difference between the corresponding values of ^, is equal and opposite to 
that along the former ; so that the circulation in the whole circuit is zero. 

If however the region be multiply-connected, an equipotential surface 
may form a barrier without dividing it into two separate parts. Let as 
many such surfaces be drawn as is possible without destroying the 
continuity of the region. The number of these cannot, by definition, be 
greater than n. Every other equipotential surface which is not closed will 
be reconcileable (in an obvious sense) with one or more of these barriers. 
A curve drawn from one side of a barrier round to the other, without meeting 
any of the remaining barriers, will cross every equipotential surface recon- 
cileable with the first barrier an odd number of times, and every other 
equipotential surface an even number of times. Hence the circulation in the 
circuit thus formed will not vanish, and <f> will be a cyclic function. 

In the method adopted above we have based the whole theory on the 

equations 

dw dv ^ ^^ ^^ _ A ^^ _ ?!f _ n (^\ 

d^^di"^' di^di" ' di'^dy'^ ' ^ ^ 

and have deduced the existence and properties of the velocity-potential in 

the various cases as necessary consequences of these. In fact, Arts. 34, 35, 

and 49, 50 may be regarded as an inquiry into the nature of the solution of 

this system of differential equations, as depending on the character of the 

region through which they hold. 

The integration of (3), when we have, on the right-hand side, instead of 
^ero, known functions of a;, y, z, will be treated in Chapter vn. 

52. Proceeding now, as in Art. 36, to the particular case of an incom- 
pressible fluid, we remark that whether <^ be cyclic or not, its first derivatives 
dift/dx, d<f}/dy, d^jdz, and therefore all the higher derivatives, are essentially 
single-valued functions, so that <f> will still satisfy the equation of continuity 

VV = 0, (1) 

or the equivalent form II ^ dS = 0, (2) 

where the surface-integration extends over the whole boundary of any 
portion of the fluid. 

The theorem (a) of Art. 40, viz. that <{> must be constant throughout the 
interior of any region at every point of which (1) is satisfied, if it be constant 
over the boundary, still holds when the region is multiply-connected. For <f>, 
being constant over the boundary, is necessarily single- valued. 

4—2 



52 Irrotational Motion [chap, hi 

The remaining theorems of Art. 40, being based on the assumption that 
the stream-lines cannot form closed curves, wiU require modification. We 
must introduce the additional condition that the circulation is to be zero in 
each circuit of the region. 

Removing this restriction, we have the theorem that the irrotational 
motion of a liquid occupying an w-ply-connected region is determinate when 
the normal velocity at every point of the boundary is prescribed, as well as 
the value of the circulation in each of the n independent and irreducible 
circuits which can be drawn in the region. For ii <f>i, <f>2 be the (cyclic) 
velocity-potentials of two motions satisfying the above conditions, then 
<f> = <f>i — (f>2 is Sk single-valued function which satisfies (1) at every point of 
the region, and makes d<f>/dn = at every point of the boundary. Hence, 
by Art. 40, <f> is constant, and the motions determined by <t>i and (f>2 are 
identical. 

The theory of multiple connectivity seems to have been first developed by Biemann* 
for spaces of two dimensions, d propas of his researches on the theory of functions of a 
complex variable, in which connection also cyclic functions, satisfying the equation 

through multiply-connected regions, present themselves. 

The bearing of the theory on Hydrodynamics and the existence in certain cases of 
many- valued velocity-potentials were first pointed out by von Helmholtzf. The subject 
of cyclic irrotational motion in multiply-connected regions was afterwards taken up and 
fully investigated by Lord Kelvin in the paper on vortex-motion already referred to;]: . 



Kelvin's Extension of Green's Theorem. 

53. It was assumed in the proof of Green's theorem that <f> and <{>' were 
both single-valued functions. If either be a cyclic function, as may be the 
case when the region to which the integrations in Art. 43 refer is multiply- 
connected, the statement of the theorem must be modified. Let us suppose, 
for instance, that <f> is cyclic ; the surface-integral on the left-hand side of 
Art. 43 (5), and the second volume-integral on the right-hand side, are then 
indeterminate, on account of the indeterminateness in the value of <f> itself. 
To remove this indeterminateness, let the barriers necessary to reduce the 
region to a simply-connected one be drawn, as explained in Art. 48. We 
may now suppose ^ to be continuous and single-valued throughout the 

* Orundlcigen fUr eine aUgemeine Theorie der Functumen einer verdnderlichen complexen 
Grosse, Qottingen, 1861 [Maihematiache Werke, Leipzig, 1876, p. 3]. Also: "Lehrs&tze aua 
der Analysia Situs," CreUe, t. liv. (1867) [Werke, p. 84], 

t CreUe, U Iv. (1868). 

X See also Kirchhoff, "Ueber die Krafte welche zwei unendlich dunne starre Ringe in einer 
fliissigkeit scheinbar auf einander ausiiben konnen," Creile, t. Ixxi. (1869) [Oea. Ahh, p. 404]. 



52-54] Extenswii of Ghreen's Theorem 53 

region thus modified; and the equation referred to will then hold, provided 
the two sides of each barrier be reckoned as part of the boundary of the 
region, and therefore included in the surface-integral on the left-hand side. 
liCt ha-i be an element of one of the barriers, k^ the cyclic constant corre- 
sponding to that barrier, d<f}'/dn the rate of variation of <f>^ in the positive 
direction of the normal to Sa^. Since, in the parts of the surface-integral 
due to the two sides of Sci, d<f}'/dn is to be taken with opposite signs, whilst 
the value of <f> on the positive side exceeds that on .the negative side 
by Ki, we get finally for the element of the integral due to Sa^, the value 
Kid<f>'ldn\ ScTi. Hence Art. 43 (5) becomes, in the altered circumstances, 

where the surface-integrations indicated on the left-hand side extend, the 
first over the original boundary of the region only, and the rest over the 
several barriers. The coefficient of any k is evidently minus the total flux 
across the corresponding barrier, in a motion of which <(>' is the velocity- 
potential. The values of ^ in the first and last terms of the equation are to 
be assigned in the manner indicated in Art. 50. 

If <f>' also be a cyclic function, having the cyclic constants ici', k^\ &c., 
then Art. 43 (6) becomes in the same way 



//*'^^+-7/s*'.+«'7/s! 



CK7o -}"••• 



Equations (1) and (2) together constitute Lord Kelvin's extension of Green's 
theorem. 

54. If ^, <f}' are both velocity-potentials of a liquid, we have 

V^ = 0, V^' = 0, (3) 

and therefore / 1 (f> ^- dS + Kijl -^ dcxi + ic , / 1 ^ (^2 + • ■ • 

= //f|«^ + .,'//gd.,+V//g«^.+ . (4) 

To obtain a physical interpretation of this theorem it is necessary to 
explain in the first place a method, imagined by Lord Kelvin, of generating 
any given cyclic irrotational motion of a hquid in a multiply-connected 
space. 



54 Irrotational Motion [chap, iri 

Let us suppose the fluid to be enclosed in a perfectly smooth and flexible 
membrane occupying the position of the boundary. Further, • let n barriers 
be drawn, as in Art. 48, so as to convert the region into a simply-connected 
one, and let their places be occupied by similar membranes, infinitely thin^ 
and destitute of inertia. The fluid being initially at rest, let each element 
of the first-mentioned membrane be suddenly moved inwards with the given 
(positive or negative) normal velocity —d<f>/dn, whilst uniform impulsive 
pressures K^p, ic^p, . . . K^p are simultaneously applied to the negative sides of 
the respective barrier-membranes. The motion generated will be characterized 
by the following properties. It will be irrotational, being generated from 
rest ; the normal velocity at every point of the original boundary will have 
the prescribed value; the values of the impulsive pressure at two adjacent 
points on opposite sides of a membrane will differ by the corresponding value 
of «:/>, and the values of the velocity-potential will therefore differ by the 
corresponding value of k; finally, the motion on one side of a barrier will be 
continuoys with that on the other. To prove the last statement we remark, 
first, that the velocities normal to the barrier at two adjacent points on 
opposite sides of it are the same, being each equal to the normal velocity of 
the adjacent portion of the membrane. Again, if P, Q be two consecutive 
points on a barrier, and if the corresponding values of ^ be on the positive 
side <ffp, <f>Q, and on the negative side <f>^p, ^'q, we have 

and therefore <f}Q — (f>p ^^<f>Q'—<f>p, 

i.e., if PQ = 8s, mds = d<f>'ld8. 

Hence the tangential velocities at two adjacent points on opposite sides of 
the barrier also agree. If then we suppose the barrier-menxbranes to be 
hquefied immediately after the impulse, we obtain the irrotational motion 
in question. 

The physical interpretation of (4), when multiplied by — />, now follows 
as in Art. 44. The values of pK are additional components of momentum, 
and those of — H d<f>/dn . da, the fluxes through the various apertures of the 
region, are the corresponding generalized velocities. 

55. If in (2) we put <l>'=<f>, and suppose ^ to be the velocity-potential 
of an incompressible fluid, we find 

The last member of this formula has a simple interpretation in terms of the 
artificial method of generating cyclic irrotational motion just explained. The 



54-56] Kinetic Energy 65 

first term has already been recognized as equal to twice the work done by ^ 
the impulsive pressure (yf> applied to every part of the original boundary of 
the fluid. Again, pK^ is the impulsive pressure applied, in the positive 
direction, to the infinitely thin massless membrane by which the place of the 
first barrier was supposed to be occupied ; so that the expression 



-»// 



p,c,.g(fo, 



denotes the work done by the impulsive forces applied to that membrane; 
and so on. Hence (5) expresses the fact that the energy of the motion is 
equal to the work done by tha whole system of impulsive forces by which we 
may suppose it generated. 

In applying (5) to the case where the fluid extends to infinity and is at 
rest there, we may replace the first term of the third member by 



-pjj(4-0^dS, (6) 



where the integration extends over the internal boundary only. The proof 
is the same as in Art. 46. When the total flux across this boundary is zero, 
this reduces to 



-<.//* 



dS (7) 



The minimum theorem of Lord Kelvin, given in Art. 45, may now be 
extended as follows: 

The irrotational motion of a liquid in a multiply-connected region has 
less kinetic energy than any other motion consistent with the same normal 
motion of the boundary and the same value of the total flux through each 
of the several independent channels of the region. 

The proof is left to the reader. 



Sources and Sinks. 

56. The analogy with the theories of Electrostatics, the Steady Flow 
of Heat, &c., may be carried further by means of the conception of sources 
and sinks. 

A 'simple source' is a point from which fluid is imagined to flow out 
uniformly in all directions. If the total flux outwards across a small closed 
surface surrounding the point be m, then m is called the 'strength* of the 
source. A negative source is called a 'sink.' The continued existence of 
a source or a sink would postulate of course a continual creation or annihi* 
lation of fluid at the point in question. 



56 IrrotcUional Motion [chap, m 

The velocity-potential at any point P, due to a simple source, in a liquid 

at rest at infinity, is 

ff>=^ml4fnr, (1) 

where r denotes the distance of P from the source. For this gives a radial 
flow from the point, and if 88 y ^r^rOy be an element of a spherical surface 
having its centre at the source, we have 

a constant, so that the equation of continuity is satisfied, and the flux 
outwards has the value appropriate to the strength of the source. 

A combination of two equal and opposite sources ± m', at a distance hs 
apart, where, in the limit, 8s is taken to be infinitely small, and m' infinitely 
great, but so that the product m'88 is finite and equal to fi (say), is called 
a 'double source' of strength /x, and the line 8^, considered as drawn in the 
direction from — m' to + m\ is called its axis. 

To find the velocity-potential at any point (a?, y, z) due to a double 
source /x situate at {x', y', z'), and having its aids in the direction ({, m, n), we 
remark that, / being any continuous function, 

fix' + 188, y' + m88y z' + nS«) -/(a?', y\ z') 

ultimately. Hence, putting/ (x', y', %') = m'/4Mr, where 

r = {(a; - x')* + (y - y'Y + (z- «')«}*. 

wefind ^^.^{i^^, + m^ + nl-)\. (2) 

_ /x cos^ 

where, in the latter form, ^ denotes the angle which the line r, considered 
as drawn from (x', y\ z') to (x, y, z), makes with the axis (Z, m, w). 

We might proceed, in a similar manner (see Art. 82), to build up sources 
of higher degrees of complexity, but the above is sufficient for our immediate 
purpose. 

Finally, we may imagine simple or double sources, instead of existing at 
isolated points, to be distributed continuously over lines, surfaces, or volumes. 

57. We can now prove that any continuous acyclic irrotational motion of 
a liquid mass may be regarded as due to a distribution of simple and double 
sources over the boundary. 



S- (4) 



56-57] Sources and Sinks 57 

This dependfl on the theorem, proved in Art. 44, that if ^, ^' be any two 
single- valued functions which satisfy V^ » 0, V^' « throughout a given 
region, then 

//*^^-//*'s«»- <») 

where the integration exteads over the whole boundary. In the present 
application, we take (^ to be the velocity-potential of the motion in question, 
and put <l>' = 1/r, the reciprocal of the distance of any point of the fluid from 
a fixed point P. 

We will first suppose that P is in the space occupied by the fluid. Since 
<f)' then becomes infinite at P, it is necessary to exclude this point from the 
region to which the formula (5) applies; this may be done by describing a 
small spherical surface about P as centre. If we now suppose SX to refer to 
this surface, and 8S to the original boimdary, the formula gives 

At the surface S we have d/dn (1/r) = — l/r*; hence if we put SS = rHuj, 
and finally make r = 0, the first integral on the left-hand becomes = — 4mf>p, 
where <^p denotes the value of <f> at P, whilst the first integral on the right 
vanishes. Hence 

^^-iim'^-iihm^- <" 

This gives the value of (f> at any point P of the fluid in terms of the 
values of <f> and 9^/dn at the boundary. Comparing with the formulae (1) 
and (2) we see that the first term is the velocity-potential due to a surface 
distribution of simple sources, with a density — d(f>/dn per unit area, whilst 
the second term is the velocity-potential of a distribution of double sources, 
with axes normal to the surface, the density being </>, It will appear from 
equation (10), below, that this is only one out of an infinite number of surface- 
distributions which will give the same value of <f> throughout the interior. 

When the fluid extends to infinity in every direction and is at rest there, 
the surface-integrals in (7) may, on a certain understanding, be taken to refer 
to the internal boundary alone. To see this, we may take as external boimdary 
an infinite sphere having the point P as centre. The corresponding part of 
the first integral in (7) vanishes, whilst that of the second is equal to (7, the 
constant value to which, as we have seen in Art. 41, (f> tends at infinity. It 
is convenient, for facility of statement, to suppose (7=0; this is legitimate 
since we may alwaj^ add an arbitrary constant to <f>. 

When the point P is external to the surface, ^' is finite throughout the 
original region, and the formula (5) gives at once 

o'-iiim^-iihm'^ '" , 



58 Irrotational Motion [chap, ni 

where, again, in the case of a liquid extending to infinity, and at rest there, 
the terms due to the infinitely distant part of the boundary may be omitted. 

58. The distribution expressed by (7) can, further, be replaced by one of 
simple sources only, or of double sources only, over the boundary. 

Let ^ be the velocity-potential of the fluid occupying a certain region, 
and let^' now denote the velocity-potential of any possible acycUc irrotational 
motion through the rest of infinite space, with the condition that^, or^', as 
the case may be, vanishes at infinity. Then, if the point P be internal to the 
first region, and therefore external to the second, we have 

where 8n, 8n' denote elements of the normal to <iS, drawn inwards to the 
first and second regions respectively, so that djdn' = — 3/3n. By addition, we 
have 

The function (j>' will be determined by the surface- values of <^' or d(f>'/dn\ 
which are as yet at our disposal. 

Let us in the first place make </>' = </> at the surface. The tangential 
velocities on the two sides of the boundary are then continuous, but the normal 
velocities are discontinuous. To assist the ideas, we may imagine a liquid to 
fill infinite space, and to be divided into two portions by an infinitely thin 
vacuous sheet within which an impulsive pressure p<f> is applied, so as to 
generate the given motion from rest. The last term of (10) disappears, so that 

^'-iiim-m^- ■■•■••<") 

that is, the motion (on either side) is that due to a surface-distribution of 
simple sources, of density 

Secondly, we may suppose that d(f>'/dn = d<f>/dn over the boundary. This 
gives continuous normal velocity, but discontinuous tangential velocity, over 
the original botmdary. The motion may in this case be imagined to be 
generated by giving the prescribed normal velocity — dif>/dn to every point 
of an infinitely thin membrane coincident in position with the boimdary. 
The first term of (10) now vanishes, and we have 



^.=^//(^-^')|,©^. (12) 



* This inTestigation was first given by Green, from the point of view of Electrostatios, 
{.c. ante p. 44. 



57-58] Stirfdce'Distrihviioiis 59 

shewing that the motion on either side may be conceived as due to a surface- 
distribution of double sources, with density 

It may be shewn that the above representations of <j> in terms of simple 
sources alone, or of double sources alone, are unique; whereas the repre- 
sentation of Art. 57 is indeterminate*. 

It is obvious that cyclic irrotational motion of a liquid c€uinot be reproduced by any 
arrangement of simple sources. It is easily seen, however, that it may be represented by 
a certain distribution of double sources over the boundary, together with a uniform distri- 
bution of double sources over each of the barriers necessary to render the region occupied 
by the fluid simply-connected. In fact, with the same notation as in Art. 53, we find 

where is the single- valued velocity-potential which obtains in the modified region, and 
<f/ is the velocity-potential of the acyclic motion which is generated in the external space 
when the proper normal velocity -d<f>/dn is given to each element ^S of a membrane 
coincident in position with the original boundary. 

Another mode of representing the irrotational motion of a liquid, whether 
cyclic or not, will present itself in the chapter on Vortex Motion. 

We here close this account of the theory of irrotational motion. The 
mathematical reader will doubtless have noticed the absence of some im- 
portant links in the chain of our propositions. For example, apart from 
physical considerations, no proof has been offered that a function </> exists 
which satisfies the conditions of Art. 36 throughout any given simply- 
connected region, and has arbitrarily prescribed values over the boundary. 
The formal proof of 'existence- theorems' of this kind is not attempted in 
the present treatise. For a review of the literature of this part of the 
subject the reader may consult the authors cited below f. 

* Cf. Larmor, "On the Mathematical Expression of the Principle of Huyghens," Proc, Lond^ 
Maih. Soc, (2) t. i. p. 1 (1903). 

t H. Burkhardt and W. F. Meyer, "Potentialtheorie,*' and A. Sommerfeld, *'Ilandwerth- 
aufgaben in der Theorie d. part. Diff.-Gleiohungen," ^tieyc. d. nuUh, Wisa, t. ii. (1900). 



CHAPTER IV 

MOTION OF A LIQUID IN TWO DIMENSIONS 

• • • 

59. If the velocities u, v be functions of Xy y only, whilst w is zero, the 
motion takes place in a series of planes parallel to xy^ and is the same in each 
of these planes. The investigation of the motion of a liquid under these 
circumstances is characterized by certain analjrtical peculiarities; and the 
solutions of several problems of great interest are readily obtained. 

Since the whole motion is known when we know that in the plane z = 0, 
we may -confine our attention to that plane. When we speak of points and 
lines drawn in it, we shall understand them to represent respectively the 
straight lines parallel to the axis of 2, and the cylindrical surfaces having 
their generating lines parallel to the axis of z, of which they are the traces. 

By the flux across any curve we shall understand the volume of fluid 
which in unit time crosses that portion of the cylindrical surface, having the 
curve as base, which is included between the planes 2; = 0, 2 = 1. 

Let Ay P be any two points in the plane xy. The flux across any two 
lines joining AP is the same, provided they can be reconciled without passing 
out of the region occupied by the moving liquid; for otherwise the space 
included between these two Unes would be gaining or losing matter. Hence 
if A be fixed, and P variable, the flux across any line AP is a function of 
the position of P. Let ^ be this function ; more precisely, let ^ denote the 
flux across AP from right to lefty as regards an observer placed on the curve, 
and looking along it from A in the direction of P. Analytically, if Z, m be 
the direction-cosines of the normal (drawn to the left) to any element S^ of 
the curve, we have 

0=1 {lu-\- mv) ds (1) 

If the region occupied by the liquid be aperiphractic (see p. 38), is 
necessarily a single- valued function, but in periphractic regions the value of ^ 
may depend on the nature of the path AP, For spaces of two dimensions, 
however, periphraxy and multiple-connectivity become the same thiQg, so that 



59] Stream-Function 61 

the properties of 0, when it is a many-valued function, in relation to the 
nature of the region occupied by the moving Uquid, may be inferred from 
Art. 50, where we have discussed the same question with regard to ^. The 
cycUc constants of ^, when the region is periphractic, are the values of the 
flux across the closed curves forming the several parts of the internal 
boundary. 

A change, say from A U> B, oi the point from which ^ is reckoned has 
merely the effect of adding a constant, viz. the flux across a line BAy to the 
value of iji ; so that we may, if we please, regard as indeterminate to the 
extent of an additive constant. 

If P move about in such a manner that the value of ^ does not alter, it 
will trace out a curve such that no fluid anywhere crosses it, i.e. a stream-Une. 
Hence the curves ^ = const, are the stream-lines, and ^ is called the 'stream- 
function.' 

If P receive an infinitesimal displacement PQ (= hy) parallel to y, the 
increment of ^ is the flux across PQ from right to left, i.e. Bt/t = — u . PQ^ or 

«=-^^ -(2) 

Again, displacing P parallel to x, we find in the same way 

«-g <») 

The existence of a function iff related to u and v in this manner might also 
have been inferred from the form which the equation of continuity takes in 
this case, viz. 

ai + a^ = ^' (*) 

which is the analjrtical condition that udy — vdx should be an exact 
difEerential*. 

The foregoing considerations apply whether the motion be rotational or 
irrotational. The formulae for the components of vorticity, given in Art. 30, 
become 

(-0. ,-«, J-S + l^; (5) 

SO that in irrotational motion we have 

a^2+ap = o (^) 

* The function ^ was introdnced in this way by Lagrange, Nouv. nUm. de VAcad. de Berlin, 
1781 [Oeuvres, t. iv. p. 720]. The kinematical interpretation is due to Rankine, "On Plane 
Water-lines in Two Dimensions," PhU. Trans, 1864 [MiaceOaneotu Scientific Papersj London, 
1881, p. 496]. 



62 Motion of a Liquid in Two Dimensions [chap, iv 

60. In what follows we confine ourselves to the case of irrotational 
motion, which is, as we have already seen, characterized by the existence, in 
addition, of a velocity-potential <^, connected with w, v by the relations 

«-!• »--| <" 

and, since we are considering the motion of incompressible fluids only, 
satisfying the equation of continuity 

aii + a^-" ^^' 

The theory of the function <^, and the relation between its properties and 
the nature of the two-dimensional space through which the irrotational 
motion holds, may be readily inferred from the corresponding theorems in 
three dimensions proved in the last chapter. The alterations, whether of 
enunciation or of proof, which are requisite to adapt these to the case of two 
dimensions are for the most part purely verbal. 

An exception, which we will briefly examine, occurs howerer in the case of the theorem 
of Art. 39 and of those which depend upon it. 

If dd be an element of the boundary of any portion of the plane xy which is occupied 
wholly' by moving liquid, and if dn be an element of the normal to ha drawn inwards, we 
have, by Art. 36, 



I 



Sn*-* (») 



the integration extending roimd the whole boundary. If this boimdary be a circle, and if 
r, ^ be polar co-ordinates referred to the centre P of this circle as origin, the last equation 
may be written 

I ^.rrfd=0, or g- I <l>de=0. 



1 /■««' 
Hence the integral ^ j (l>d$. 



i.e. the mean value of </> over a circle of centre P and radius r, is independent of the value 
of r, and therefore remains unaltered when r is diminished without Umit, in which case it 
becomes the value of <f) at P. 

If the region occupied by the fluid be periphractio, and if we apply (3) to the space 
enclosed between one of the internal boundaries and a circle with centre P and radius r 
surrounding this boundary, and lying wholly in the fluid, we have 



where the integration in the flrst member extends over the circle only, and M denotes the 
flux into the region across the internal boundary. Hence 

M 



i: 



""^.rde^-Mi (4) 



^'kiy^^^' 



• 1 fi^ M 
which gives on integration ^ \ ^(W = -5- log r + 0; (5) 

i.e. the mean value of <f> over a circle with centre P and radius r is equal to - if /2tr . log f-¥Cy 



60-61J Khietic Energy 63 

where C is independent of r but may vary with the position of P. This formula holds of 
course only so far as the circle embraces the same internal boundary, and lies itself wholly 
in the fluid. 

If the region be unlimited eztemaUy, and if the circle embrace the whole of the 
internal boundaries, and if further the velocity be everywhere zero at infinity, then C 
is an absolute constant; as is seen by reasoning similar to that of Art. 41. It may then 
be shewn that the value of at a very great distance r from the internal boundary tends 
to the value - MI2rr . log r + 0. In the particular case of j9f =0 the limit to which tends 
at infinity is finite; in all other cases it is infinite, and of the opposite sign to M. 
We infer, as before, that there is only one single- valued function <f> which satisfies the 
equation (2) at every point of the plane xy external to a given system of closed curves, 
makes the value of d<^/3n equal to an arbitrarily given quantity at every point of 
these curves, and has its first differential coefficients all zero at infinity. 

If we imagine point-sources, of the type explained in Art. 56, to be distributed uni- 
formly along the axis of 2, it is readily found that the velocity at a distance r from this 
axis will be in the direction of r, and equal to m/2rrr, where m is a certain constant. This 
arrangement constitutes what may be called a * line-source ,* and its velocity-potential may 
be taken to be 

*=-^log»' -(6) 

The reader who is interested in the matter will have no difficulty in working out a theory 
of two-dimensional sources and sinks, similar to that of Arts. 56 — 58*. 

61. The kinetic energy T of a portion of fluid bounded by a cylindrical 
surface whose generating lines are parallel to the axis of z, and by two 
planes perpendicular to the axis of z at unit distance apart, is given by the 
formula 

^^-'//{(i)"-(^)>*--/*l* '" 

where the surface-integral is taken over the portion of the plane xy cut off 
by the cylindrical surface, and the line-integral round the boundary of this 
portion. Since d<f>/dn = — dift/dsy the formula (1) may be written 

2r = />J^#, (2) 

the integration being carried in the positive direction round the boundary. 

If we attempt by a process similar to that of Art. 46 to calculate the energy in the case 
where the region extends to infinity, we find that its value is infinite, except when the total 
flux outwards (Jf ) is zero. For if we introduce a circle of great radius r as the external 
boundary of the portion of the plane xy considered, we find that the corresponding part 
of the integral on the right-hand side of (1) increases indefinitely with r. The only 
exception is when Jf =0, in which case we may suppose the line-integral in (1) to extend 
over the internal boundary only. 

If the cylindrical part of the boundary consist of two or more 
separate portions one of which embraces all the rest, the enclosed region 

* This subject has been treated very fully by C. Neumann, UAer das logarithmisehe und 
NewUm'ache Potential, Leipsig, 1877. 



64 Motion of a Liquid in Two Dimensioiis [chap, iv 

is multiply-connected, and the equation (1) needs a correction, which may 
be applied exactly as in Art. 55. 

62. The functions <^ and ^ are connected by the relations 

d<f> __d^ d<f> ^ dtff . 

dx~^ dy* dy ~~ dx ^ ^ 

These conditions are fulfilled by equating ^ + 10, where i stands as usual 
for \/( — 1), to any ordinary algebraic or transcendental function of x + iy, say 

<t> + i^=f{x-hiy) (2) 

For then |^ (<^ + i^) = if {x + iy) = *" ^ (^ + **0), (3) 

whence, equating separately the real and the imaginary parts, we see that 
the equations (1) are satisfied. 

Hence any assumption of the form (2) gives a possible case of irrotational 
motion. The curves </> = const, are the curves of equal velocity^potential, 
and the curves ^ = const, are the stream-lines. Since, by (1), 

d<f> dtjt d<f>d^ ^ ^ 
dx dx dy dy ' 

we see that these two systems of curves cut one another at right angles, as 
already proved. Since the relations (1) are unaltered when we write — \ft for 
<f>, and (f> for 0, we may, if we choose, look upon the curves ^ = const, as the 
equipo'tential curves, and the curves <l> = const, as the stream-lines ; so that 
every assumption of the kind indicated gives us two possible cases of 
irrotational motion. 

For shortness, we shall through the rest of this chapter follow the usual 
notation of the Theory of Functions, and write 

z^x-^-iy, (4) 

w = <f) -{- iff/ (5) 

From a modem point of view, the fundamental property of a function 
of a complex variable is that it has a definite differential coefiicient with 
respect to that variable*. Tf ^, iff denote any functions whatever of x and y, 
then corresponding to every value of a; 4- iy there must be one or more 
definite values of ^ + iif/ ; but the ratio of the differential of this function 
to that oi x + iy, viz. 

8a: + iSy * hx + r 8y ' 

* See, for example, Forsyth, Theory of Functions^ 2nd ed., Cambridge, 1900, cc. i., ii. 



61-62] Complex Variable 66 

depends in general on the ratio 8x : hy. The condition that it should be the 
same for all values of the latter ratio is 

|+'|-'(S+'D ■■<'" 

which is equivalent to (1) above. This property was adopted by Biemann 
as the definition of a function of the complex variable x-^ iy\ viz. such 
a function must have, for every assigned value of the variable, not only a 
definite value or system of values, but also for each of these values a definite 
differential coefficient. The advantage of this definition is that it is quite 
independent of the existence of an analytical expression for the function. 

If the complex quantities z and w be represented geometrically after 
the manner of Argand and Gauss, the differential coefficient dwjdz may be 
interpreted as the operator which transforms an infinitesimal vector hz into 
the corresponding vector hw. It follows then, from the above property^ that 
corresponding figures in the planes of z and w are similar in their infinitely 
small parts. 

For instance, in the plane of w the straight lines ^ = const., ^ = const., 
where the constants have assigned to them a series of values in arithmetical 
progression, the common difference being infinitesimal and the same in each 
case, form two systems of straight lines at right angles, dividing the plane 
into infinitely small squares. Hence in the plane xy the corresponding 
curves ^ = const., ^ = const., the values of the constants being assigned as 
before, cut one another at right angles (as has already been proved otherwise) 
and divide the plane into infinitely small squares. 

Conversely, if ^, '^ be any two functions of ar, y such that the curves ^=mc, yft=n€, 
where c is infinitesimal, and m, n are any integers, divide the plane xy into elementary 
squares, it is evident geometrically that 

If we take the upper signs, these are the conditions that x +iy should be a function of 
<f) +%ylt. The case of the lower signs is reduced to this by reversing the sign of ^. Hence 
the equation (2) contains the complete solution of the problem of orthomorphic projection 
from one plane to another*. 

The similarity of corresponding infinitely small portions of the planes w 
and z breaks down at points where the differential coefficient dw/dz is zero 
or infinite. Since 

t't+'t (') 

the corresponding value of the velocity, in the hydrodynamical application, 
is zero or infinite. 

* Lagrange, "Sur la construction des cartes g^graphiques/* Nouv, nUm, de VAcad. de Berlin, 
1779 [Oeiivres, t. iv. p. 636]. For the further history of the problem, see Forsyth, Theory of 
Functions, c. xix. , 

L. H. 5 



(1) 



66 Motion of a Liquid in Two Dimensions [chap, iv 

In all. physical applications, w must be a single- valued, or at most 
a cyclic function of 2, in the sense of Art. 50, throughout the region 
with which we are concerned. Hence in the case of a 'multiform' function, 
this region must be confined to a single sheet of the corresponding Biemann's 
surface, and * branch-points* therefore must not occur in its interior. 

63. We can now proceed to some applications of the foregoing method. 

First Jet us assume w = Az^, 

A being real. Introducing polar co-ordinates r, d, we have 

if, = Ar^ cos ndy ' 
^ = Ar^ Binnd. ) 

The following cases may be noticed. 

1^. If n = 1, the stream-lines are a system of straight lines parallel to x, 
and the equipotential curves are a similar system parallel to y. In this case 
any corresponding figures in the planes of w and z are similar, whether they 
be finite or infinitesimal. 

2^. If n= 2, the curves </> = const, are a system of rectangular hyperbolas 
having the axes of co-ordinates as their principal axes, and the curves 
^ = const, are a similar system, having the co-ordinate axes as asymptotes. 
The lines 6 = 0, = ^tt are parts of the same stream-line ^ = 0, so that we 
may take the positive parts of the axes of x, y as fixed boundaries, and thus 
obtain the case of a fluid in motion in the angle between two perpendicular 
walls. 

3°. If n = — 1, we get two systems of circles touching the axes of 
co-ordinates at the origin. Since now <f) = A/r . cos 0, the velocity at the 
origin is infinite ; we must therefore suppose the region to which our formulae 
apply to be limited internally by a closed curve. 

4°. If n = — 2, each system of curves is composed of a double system of 
lemniscates. The axes of the system <f> = const, coincide with a; or y ; those 
of the system = const, bisect the angles between these axes. 

5°. By properly choosing the value of w we get a case of irrotational 
motion in which the botmdary is composed of two rigid walls inclined at any 
angle a. The equation of the stream-lines. being 

T^BinnO = const., (2) 

we see that the lines = 0, = ir/n are parts of the same stream-line. 
Hence if we put n = irja, we obtain the required solution in the form 

<f) = Ar^ cos — , Jt = Ar^ sin — (3) 

a a 



62-64] Examples 67 

The component velocities along and perpendicular to r are 

— il-f COS — , and A-r sm — , 
a <x a a 

and are therefore zero, finite, or infinite at the origin, according as a is less 
than, equal to, or greater than tt. 

64. We take next some cases of cyclic functions. 

1°. The assumption «r = — /x log 2, (1) 

where /i is real, gives ^ = — ft log r, = — /xfl (2) 

The velocity at a distance r from the origin is /i/r ; this point must therefore 
be isolated by drawing a closed curve round it. 

If we take the radii 6 = const, as the stream-lines we get the case of 
a (two-dimensional) source at the origin. (See Art. 60.) 

If the circles r = const, be taken as stream-lines we have the case of 
Art. 27 ; the motion is now cycUc, the circulation in any circuit embracing 
the origin being 27rfi. 

2°. Let us take w = — u log — ; — (3) 

If we denote by r^, r^ the distances of any point in the plane xy from the 
points (± a, 0), and by 0^, ^2 ^^^ angles which these distances make with 
the positive direction of the axis of a;, we have 

whence ^ = — ft log fi/r,, ^ = — ft (^i — B^ (4) 

The curves ^ = const., ^ == const, form two orthogonal sjrstems of 'coaxal' 
circles ; see p. 68. 

Either of these systems may be taken as the equipotential curves, and 
the other system will then form the stream-Unes. In either case the velocity 
at the points (± a, 0) will be infinite. If these points be accordingly isolated 
by drawing closed curves round them, the rest of the plane xy becomes 
a triply-connected region. 

If the circles 6^ — 62"= const, be taken as the stream-lines we have the 
case of a source and a sink, of equal intensities, situate at the points (± a, 0). 
If a is diminished indefinitely, whilst iia remains finite, we reproduce the 
assumption of Art. 63, 3^, which therefore corresponds to the case of a double 
line-source at the origin. (See the first diagram of Art. 68.) 

If, on the other hand, we take the circles r-^r^ = const, as the stream-lines 
we get a case of cyclic motion, viz. the circulation in any circuit embracing 
the first (only) of the above points is 27rft, that in a circuit embracing the 

6—2 



68 



Motion of a Liquid in Two Dimensions [chap, iv 



second is — 27r/x ; whilst that in a circuit embracing both is zero. This 
example will have additional interest for us when in Chapter vii. we come 
to treat of 'Kectilinear Vortices.' 




3°. The potential- and stream-functions due to a row of equal and 
equidistant sources at the points (0, 0), (0, ± a), (0, ± 2a), . . . are given by 
the formula 

u; oclog 2 4- log (z — ia) 4- log (2 4- io) 4- log {z — 2ia) 4- log (z 4- 2ia) 4- . . . , 

(5) 

ttz 
or, say, w = C log sinh — , (6) 



a 



where C is real. This makes 



in agreement with a result given by Maxwell*. The formulae apply also to 
the case of a source midway between two fixed boundaries y = ± ^a. 

The case of a row of double sources having their axes parallel to x is 
obtained by differentiating (6) with respect to z. Omitting a factor we have 



w = Ccoth — , 

a 



(8) 



^j . _ C sinh (27ng/a) ,^ C sin (27Ty/g) 

^ cosh {^TTx/a) — cos (27ry/a) ' ^ cosh (^ttx/o) — cos (iiTy/a) ' "^ ^ 



* Electricity and Magnetism, Art. 203. 



64-65] Inverse Formulae 69 

Superposing a uniform motion parallel to x negative, we have 

w? = 2 + C coth ^ , (10) 

a 

or 

, __ C sinh (iirxja) . _ C sin {^myja) 

^ cosh {27rx/a) — cos (^Try/a) ' ^ ~ ^r ^^gj^ ^27Tx/a) — cos (^Try/a) ' 

(11) 

The stream-line ^ = now consists in part of the line y = 0, and in part of 
an oval curve whose semi-diameters parallel to x and y are given by the 
equations 

sinh«!!? = ![2, ytan^ = C (12) 

If we put C = Trb^a, (13) 

where 6 is small compared with a*, these semi-diameters are each equal to 
i, approximately. We thus obtain the potential- and stream-functions for 
a liquid flowing through a grating of parallel cylindrical bars of small circular 
section. The second of equations (11) becomes in fact, for small values of x, y^ 



^=yi^-^^ (1^) 



x^ + y\ 

65. If ti; be a function of z, it follows at once from the definition of 
Art. 62 that 2; is a function of w. The latter form of assumption is some- 
times more convenient analytically than the former. 

The relations (1) of Art. 62 are then replaced by 

dx_dy dx__dy ,,. 

d4>'^d^' difs" d<f> ^^ 

Also smce -^- = ^-1-^^ = — w + *v, 

, dz 1 1 /u , ,v\ 

we have — — - = == _ ( _ 4- ^ - 

dw u — IV q\q qj 
where q is the resultant velocity at (ar, y). Hence if we write 

i—t <2> 

and imagine the properties of the function ^ to be exhibited graphically in 
the manner already explained, the vector drawn from the origin to any point 
in the plane of t, will agree in direction with, and be in magnitude the 
reciprocal of, the velocity at the corresponding point of the plane of z. 

* The approzimately ciroular form holds however for a considerable range of values of C. 
Thus if we put C=ia, we find from (12) 

a:/a = -264, y fa = -250. 

The two diameters are very nearly equal, although the breadth of the oval is half the interval 
between the stream-lines y = ± Ja. 



70 Motion of a Liquid in Two Dimensions [chap, iv 

Again, since 1/g is the modulus of dzjdwy i.e. of dxjd^ + idy/d<f>, we have 

h-&'H%)' <'> 

which may, by (1), be put into the equivalent forms 

(4) 

The last formula, viz. -z = .A-r-iiy (5) 

expresses the fact that corresponding elementary areas in the planes of z 
and w are in the ratio of the square of the modulus of dz/dw to unity. 

66. The following examples of this procedure are important. 

l^ Assume z — ccoshw, (1) 

or a? = c cosh <f> cos ^,1 .^. 

y = csinh^sin 0.) 

The curves <f> = const, are the ellipses 

_^!_ + __lL__i (3) 

c» co8h« <f,^e* sinh* <f>~ ' ^ ' 

and the curves tfi = const, are the hyperbolas 

^' t 1 (4) 

c* COS* ^ c* sin* ^ ' 

these conies having the common foci (± c, 0). The two systems of curves are 
shewn on the opposite page. 

Since at the foci we have <f> = 0, ff/ = nir, n being some integer, we see by 
(2) of the preceding Art. that the velocity there is infinite. If the hyperbolas 
be taken as the stream-lines, the portions of the axis of x which lie outside 
the points (± c, 0) may be taken as rigid boundaries. We obtain in this 
manner the case of a liquid flowing from one side to the other of a thin plane 
partition, through an aperture of breadth 2c; the velocity at the edges is 
however infinite. 

If the ellipses be taken as the stream-lines we get the case of a liquid 
circulating round an elliptic cylinder, or, as an extreme case, round a rigid 
lamina whose section is the line joining the foci {± c, 0). 

At an infinite distance from the origin <f} is infinite, of the order Togr, 
where r is the radius vector; and the velocity is infinitely small of the 
order 1/r. 



65-66] 



Examples 



71 




2^ Let 2 = w + e^ (5) 

or x = <f> + e* cos ^, y = ^ + c* sin ^ (6) 

« 

The stream-line ^ = coincides with the axis of x. Again, the portion of 
the line y = tt between x = — oo and a; = — 1, considered as a line bent back 
on itself, forms the stream-line ^ = tt ; viz. as ^ decreases from + oo through 
to — 00 , X increases from — oo to — 1 and then decreases to — oo again. 
Similarly for the stream-line ^ = — tt. 

Since il = — dz/dw = — 1 ~ e* cos ^ — le* sin ^, 

it appears that for large negative values of <f> the velocity is in the direction 
of x-negative, and equal to unity, whilst for large positive values it is zero. 

The above formulae therefore express the motion of a liquid flowing into 
a canal bounded by two thin parallel walls from an open space. At the ends 
of the walls we have ^ = 0, ^ = ± tt, and therefore ? = 0, i.e. the velocity is 
infinite. The direction of the flow will be reversed if we change the sign of 
w in (5). The forms of the stream-lines, drawn, as in aU similar cases in this 
chapter, for equidistant values of ^, are shewn on the next page*. 

* This example was given by Helmholtz, Berl MonaUber. April 23, 1868 [PhU, Mag. 
Nov. 1868, Wise, Ahh, t. i. p. 164]. 



72 



Motion of a Liquid in Two Dimensions [chap, iv 



If the walls instead of being parallel make angles ± p with the line of 
symmetry, the appropriate formula is 



z = 



n 



(1 __ g-n») ^ g(l-n)w 



(7) 



where n = p/ir. The stream-lines ^ = ± tt follow the course of the walls*. 
This agrees with (5) when n tends to the limit 0, whilst if n = J we have 
virtually the case shewn on the preceding page. 





67. It is known that a function / (z) which is finite, continuous, and 
single-valued, and has its first derivative finite, at all points of the space 
included between two concentric circles about the origin, can be expanded 
in the form 

f{z) = Ao + A^z 4- A^z^ 4- . . . + B^z-^ + B^z-^ + (1) 

If the above conditions be satisfied at all points within a circle having the 
origin as centre, we retain only the ascending series ; if at all points without 
such a circle, the descending series, with the addition of the constant A^^ is 
sufficient. If the conditions be fulfilled for all points of the plane ani without 
exception,/ (2) can be no other than a constant A^. 

* R. A. Harris, "On Two-Dimendonal fluid Motion through Spouts composed of two Plane 
WaUs,'* Ann. of Math, (2), t. ii. (1901). A diagram is given for the case of /3=jtx. 



66-68] General Formvlae 73 

Putting / (2) =^ + i^, introducing polar co-ordinates, and writing the 
complex constants A^ , B^ in the forms P^ 4- iQ^ , R^ + iS^ , respectively, 
we obtain 

^ = Po + 2rr«(P„cosne-gn8inne) + 2rr-«(fi„cosne + /S„sinne),) 
= Qo + ^r^** (On cos n^ + P„ sin nJd) + S^f-" (iS„ cos nfi - fi„ sin wfl) J 

These formulae are convenient in treating problems where we have the 
value of <f>, or of d<f>ldn^ given over the circular boundaries. This value may 
be expanded for each boundary in a series of sines and cosines of multiples 
of 0, by Fourier's theorem. The series thus found must be equivalent to 
those obtained from (2) ; whence, equating separately coefficients of sin r\d 
and cosnd, we obtain equations to determine P„, Q„, 22^, iS^. 

68. As a simple example let us take the case o f an infinitely long circular y^/'- ^^^ 
cylinder of radius a moving with velocity U perpendicular to its length, in an ^ 
infinite mass of liquid which is at rest at infinity. 

Let the origin be taken in the axis of the cylinder, and the axes of x, y 
in a plane perpendicular to its length. Further let the axis of x be in the 
direction of the velocity t7. The motion, supposed originated from rest, will 
necessarily be irrotational, and <f> will be single- valued. Also, since jd<f>/dn . ds, 
taken round the section of the cylinder, is zero, ^ is also single-valued 
•(Art. 59), so that the formulae (2) apply. Moreover, since 3^/9n is given at 
every point of the internal boundary of the fluid, viz. 

- ^ = t7 cos «, f or r = a, (3) 

and since the fluid is at rest at infinity, the problem is determinate, by 
Art. 41. These conditions give P„ = 0, Qn = 0, and 

U cos ^ = 2" na-^'^ (22„ cos nB -h S^sm nd), 

which only c^n be satisfied by making Ri = Ua\ and all the other coefficients 
zero. The complete solution is therefore 

<f> = — cos 6, ^ = BinO (4) 

The stream-Unes ^ = const, are circles, as shewn on the next page. 

The kinetic energy of the liquid is given by the formula (2) of Art. 61, viz. 

2T = pUd^ = pU^^ j^coB^ede = M'U\ (5) 

if ibf ', = ^na^p, be the mass of fluid displaced by unit length of the cylinder. 
This result shews that the whole effect of the presence of the fluid may b e 
represe nted by an addition M' to the inertia per umt le ngth of the cyhnde r. 



74 



Motion of a Liquid in Two Dimendons [chap, iy 



Thus, in the case of lectilinear motion, if we have an extraneous force X per 
unit length acting on the cylindei, the equation of energy gives 



or 



{M + M')^ = X, 



dt 



(6) 



where M represents the mass of the cylinder itself. 
Writing this in the form 

dt dt' 

we learn that the pressure of the fluid is equivalent to a force — M'dUjdt 
per unit length in the direction of motion. This vanishes when TJ is constant. 




The above result must of course admit of verification by direct calculation. By 
Art. 20 (6) the pressure is given by the formula 

M-fe'-^c) (') 

provided q denote the velocity of the fluid relative to the centre of the moving sphere. 
The term due to the extraneous forces (if any) acting on the fluid has been omitted ; the 
effect of these would be given by the rules of Hydrostatics. We have, for r =a, 

^ = a^cosd, g«=4C^«sin«d, (8) 

whence p =p (a -^- cos 6 -2U^ sin« 6 + J^ (O) (9) 



68-69] 



Motion of a Circular Cylinder 



75 



The resultant fcxrce on unit length of the cylinder is evidently parallel to the initial line 
^=0; to find its amount we multiply by -adB , cos B and integrate with respect to 6 
between the limits and 2v, The result is -M'dU/dt^ as before. 

If in the above example we impress on the fluid and the cylinder a 
velocity — t7 we have the case of a current flowing with the general velocity 
U past a fixed cylindrical obstacle. Adding to <f> and tft the terms Ur cos 
and Ur sin 0, respectively, we get 



<f>=u(r + ^cod0, ^^U(r-^)sm0. 



(10) 



If no extraneous forces act, and if {7 be constant, the resultant force on the 
cylinder is zero. Cf. Art. 92. 




69. To render the formula (1) of Art. 67 capable of representing any 
case of continuous irrotational motion in the space between two concentric 
circles, we must add to the right-hand side the term 

^log^ (1) 

li A ^ P -i- iQ, the corresponding terms in ^, ^ are 

P]osr^Q0, P0-{-Qlogr ......(2) 

respectively. The meaning of these terms is evident; thus 2nPy the cycUc 
constant of iff, is the flux across the inner (or outer) circle; and 27tQ, the 
cyclic constant of <f>, is the circulation in any circuit embracing the origin. 

For example, returning to the problem of the last Art., let us suppose that 
in addition to the motion produced by the cylinder we have an independent 



76 



Motion of a Liquid in Two Dimensions [chap, iv 



circulation round it, the cyclic constant being k. The boundary-condition is 
then satisfied by 



a' 



K 



^ r 27T 



(3) 



The effect of the cyclic motion, superposed on that due to the cylinder, 
will be to augment the velocity on one side, and to diminish (and, it may be, 
to reverse) it on the other. Hence when the cylinder moves in a straight 
line with constant velocity, there will be a diminished pressure on one side, 
and an increased pressure on the other, so that a constraining force must be 
applied at right angles to the direction of motion. 




The figure shows the lines of flow. At a distance from the origin they approximate to 
the form of concentric circles, the disturbance due to the cylinder becoming small in com- 
parison with the cyclic motion. When, as in the case represented, U > ic/2ira, there is a 
point of zero velocity in the fluid. The stream-line system has the same configuration in 
all cases, the only effect of a change in the value of U being to alter the scale, relative to 
the diameter of the cylinder. 

When the problem is reduced to one of steady motion we have in place of (3) 



<f> = v(r.^) 



COS 6 -^r-B, 



(4) 



whence 



-= const. -\if 
P 



= const 



•-l(: 



2Usine^.^y 



(5) 



69] Cylinder with Circulation 77 



/: 



for r=a. The resultant pressure on the cylinder is therefore 

Sir 

I? sin Sad6 = -KpU, (6) 



at right angles to the general direction of the stream. This result is independent of the 
radius of the cylinder. It is not difficult indeed to shew, with the help of principles 
developed later in this treatise, that it holds for any form of section*. 

To calculate the effect of the fluid pressures on the cylinder when moving in any 
manner we may conveniently adopt moving axes, the origin being taken at the centre, 
and the axis of x in the direction of the velocity U. If ;( be the angle which this makes 
with a fixed direction, the equation (6) of Art. 20 gives 



p~ dt ^ dt de' 



i^'-tfe (^) 

where q now denotes fluid velocity relative to the origin, to be calculated from the relative 
velocity-potential <f> + Ur cos 3, <t> being given by (3). We find, for r=a, 

? = «fcoe^-i(2t;sin^+4)%aC;|Bin^+JLj («) 

The resultant pressures parallel to x and y are therefore 

- (^'pcoBBad0='M'^, - l^' pHinBadO^^KpU -M'U^ (9) 

Jo at J <U 

where M' = irpa* as before. 

Hence if P, Q denote the components of the extraneous forces, if any, acting on the 
cylinder in the directions of the tangent and the normal to the path, respectively, the 
equations of motion of the cylinder are 

\ (10) 

If there be no extraneous forces, U is constant, and writing dxidi = VIR, where R is the 
radius of curvature of the path, we find 

R = (M+M') U/kp (11) 

The path is therefore a circle, described in the direction of the cyclic motion f. 

liifTfhe the rectangular co-ordinates of the axis of the cylinder, the equations ( 10) aro 
equivalent to 

{M+M')ij= kp( + yJ 

where X, Y are the components of the extraneous forces. To find the effect of a constant 
force, we may put 

X={M+M')g\ y=0 (13) 

The solution then is f = a + c cos (ni + c), 



(14) 



ri=p + - <+csin(n«+€), 

where a, /3, c, r are arbitrary constants, and 

n = Kpl(M +3f' ) (15) 

* This remark is due to Kutta and Joukowaki; see Kutta, Sitzb. d. k, hayr, Akad. d, Wias. 
1910. 

t Rayleigh, "On the Irregular Flight of a Tennis Ball," Mesa, of Math. t. vii. (187S) [Papers, 
t. i. p. 344]; Greenhill, Mess, of Math. t. iz. p. 113 (1880). 



78 Motion of a Liquid in Two Dimensions [chap, iv 

This shews that the path is a troohoid, deeoribed with a mean velocity g'jn perpendicular 
to a;*. It is remarkable that the cylinder has on the whole no progressive motion in the 
direction of the extraneous force. In the particular case c =0 its path is a straight line 
perpendicular to the force. The problem is an illustration of the theory of 'gyrostatic 
systems,* to be referred to in Chapter vi. 

70. The formula (I) of Art. 67, as amended by the addition of the term 
A log z, may readily be generalized so as to apply to any case of irrotational 
motion in a region with circular boundaries, one of which encloses all the 
rest. In fact, for each internal boundary we have a series of the form 

41og(«-c) + -^'^ + ^-^,+ .... 

where c,^a-^ib say, refers to the centre, and the coefficients A^ A^, A^, ... 
are in general complex quantities. The difficulty however of determining 
these coefficients so as to satisfy given boundary conditions is now so great 
as to render this method of very limited application. 

Indeed the determination of the irrotational motion of a liquid subject to 
given boundary conditions is a problem whose exact solution can be effected 
by direct processes in only a very few cases f. Most of the cases for which we 
know the solution have been obtained by an inverse process ; viz. instead of 
tr}ring to find a value of ^ or ^ which satisfies V\f> » or V^ = and given 
boundary conditions, we take some known solution of the differential equations 
and enquire what boundary conditions it can be made to satisfy. Examples 
of this method have already been given in Arts. 63, 64, and we may carry it 
further in the following two important cases of the general problem in two 
dimensions. 

71. Case I. The boundary of the fluid consists of a rigid cylindrical 
surface which is in motion with velocity 17 in a direction perpendicular to its 
length. 

Let us take as axis of x the direction of this velocity Z7, and let S^ be an 
element of the section of the surface by the plane xy. 

Then at all points of this section the velocity of the fluid in the direction 
of the normal, which is denoted by ^;?^» must be equal to the velocity of 

* GMQhkll It. 

t A T«ry pow«ffal melhod of transfomiatioii, applkable to cases where the boundaries of 
the fluid conaisl of iixMl plane walls, has however been developed by Schwajrx, "Ueber einige 
Abbildiin^satt^bMU** OcUt, l« Ixx. [CawiittiMlfo J6Aaii4liiiife», Berlin. 1890» t. iL il 65]; 
€1iristoffel» ^^'Snl problems deQe tempeialnre stazionarie e la rapptesentaiione di una data 
saperfici^'* Anmsiii di M^Ummtiea (2), t. i pw S9» and Kirchhoff, '^Zor Tbeorie des Oandensalocs^" 
BvL Mo v ^ hUr. March 15, 1877 \iU^ Ahk. p^ 101]. Many of the aolntians which can be thus 
obtained are of great interr^ in the mathematieaUj cognate snbjects of Ekctrostaties, Heat- 
Oondiaelioa, Jbw $ee« for example, J. J. Thomson* Reemi Resemrckes • a EUctncUif «W Jfaynetfim. 
Oxioid. l^SO, e. iii 



6&-71] Inverse Methods 79 

the boundary normal to itself, or — TJdylda. Integrating along the section, 
we have 

^= — Uy + const (1) 

If we take any admissible form of ^, this equation defines a system of curves 
each of which woidd by its motion parallel to x give rise to the stream-lines 
^ = const.* We give a few examples. 

1°. If we choose for iff the form — Uy^ (1) is satisfied identically for all 
forms of the boundary. Hence the fluid contained within a cylinder of any 
shape which has a motion of translation only may move as a solid body. 
If, further, the cyUndrical space occupied by the fluid be simply-connected, 
this is the only kind of motion possible. This is otherwise evident from 
Art. 40; for the motion of the fluid and the solid as one mass evidently 
satisfies all the conditions, and is therefore the only solution which the problem 
admits of. 

2°. Let ^ = A/r . sin &; then (I) becomes 

— sin ^ = — Ur sin d + const (2) 

r ^ ' 

In this system of curves is included a circle of radius a, provided A/a = — Ua. 
Hence the motion produced in an infinite mass of liquid by a circidar cylinder 
moving through it with velocity u perpendicular to its length, is given by 

^=-^%infl, (3) 

which agrees with Art. 68. 

3°. Let us introduce the elliptic co-ordinates ^, rj, connected with x, y 

by the relation 

x-\' iy = c cosh (f -f iiy), (4) 

or x = c cosh ^ cos 97,1 .^. 

y = csinh ^sinij, j 

(cf . Art. 66), where f may be supposed to range from to 00 , and 17 from 
to 27r. If we now put 

^ 4- i^ = Ce-(^+^''>, (6) 

where G is some real constant, we have 

^ = - Ce-f siniy, (7) 

so that (1) becomes Ce~* sin 17 == t7c sinh ^ sin 17 -}- const. 

In this system of curves is included the ellipse whose parameter ^0 ^ 
determined by 

Ce^ = Uc sinh ^0 • 

* Cf. Rankine, 2.c. ante p. 61, where the method is applied to obtain curves resembling the 
lines of shipSi 



80 



Motion of a Liquid in Two Dimensions [chap, iv 



If a, b be the semi-axes of this ellipse we have 

a = c cosh ^o> b = c sinh ^q, 
Ubc __ jj. fa + b\\ 



so that 



C = 



a — b \a — 6 

a 4- b\\ 



7j e^^sin^y (8) 

gives the motion of an infinite mass of liquid produced by an elliptic 
cylinder of semi-axes a, 6, moving parallel to the greater axis with velocity TJ , 

That the above formulae make the velocity zero at infinity appears from 
the consideration that, when ^ is large, hx and 8y are of the same order as 
e^S^ or c^Sjy, so that dtff/dx, hffjdy are of the order e~^ or 1/r*, ultimately, 
where r denotes the distance of any point from the axis of the cylinder. 

If the motion of the cylinder were parallel to the minor axis, the formula 
would be 

^=^«(— ft)* «-*<«» ^ (9) 




The stream-lines are in each case the same for all confocal elliptic forms 
of the cylinder, so that the formulae hold even when the section reduces to 
the straight line joining the foci. In this case (9) becomes 

^ = Fc c-f cos ly, (10) 






71] Translation of an Elliptic Ct/linder 81 

which would give the motion produced by an infinitely long lamina of 
breadth 2c moving 'broadside on' in an infinite mass of liquid. Since 
however this solution makes the velocity infinite at the edges, it is subject 
to the practical Umitation already indicated in several instances*. 

The kinetic energy of the fluid is given by 

2T = pj<f>cklf = pCh'^^* I "" C08« rfdrj 

=^7rph^V\ (11) 

where h is the half-breadth of the cylinder perpendicular to the direction of 
motion. 

If the units of length and time be properly chosen we may write 

X +iy =co8h (( +iff), <t> +»> =6" ^^■^•''^ 

whence ^ = ,^ (n.^._L_) , y=^(i__L_). 

These formulae are convenient for tracing the curves (^= const.. ^= const., which are 
figured on the preceding page. 

By superposition of the results (8) and (9) we obtain, for the case of an elliptic cylinder 
having a velocity of translation whose components are {7, F, 

Vr = - (^)**"^(^^ sin 17 - Fa cos ly) (12) 

To find the motion relative to the cylinder we must add to this the expression 

Uy- Fa;=c(f7 8inhf siniy- Fcosh f cos 17) ^ (13) 

For example, the stream-function for a current impinging at an angle of 45° on a plane 
lamina whose edges are at a? = ± c is 

V^ = -;y2 ^0^ ^^ ^ (*^^ ^ -fidni?), .(14) 




* This investigation was given in the Quart. Joum, of McUh, t. xiv. (1875). Results 
equivalent to (8), (9) had however been obtained, in a different manner, by Beltrami, "Sui 
prinoipii fondamentali dell' idrodinamica razionale," Mem, delV Accad, deUe Scienze di Bologna^ 
1873, p. 394. 

L. H. 6 



82 Motion of a Liquid in Two Dimensions [chap, iv 

where q^ is the velocity at infinity. This immediately verifies, for it makes ^ =0 for f =0, 
and gives 

V'=-||(x-y) 

for ^ = 00 . The stream-lines for this case djce shewn in the annexed figure (turned through 
45° for convenience)*^. This will serve to illustrate some results to be obtained later in 
Chapter VL 

If we trace the course of the stream-hue -^ =0 from <^=+oo to<^=-oo,we find that it 
consists in the first place of the hyperbolic arc i; -\rr, meeting the lamina at right angles; 
it then divides into two portions, following the faces of the lamina, which finally re-unite 
and are continued as the hyperbolic arc i7=jir. The points where the hyperbolic arcs 
abut on the lamina dse points of zero velocity, and therefore of maximum pressure. It is 
plain that the fluid pressures on the lamina are equivalent to a couple tending to set it 
broadside on to the stream; and it is easily found that the moment of this couple, per 
unit length, is \irpq^c\ Compare Art. 124t. 

72. Casb II. The boundary of the fluid consists of a rigid cylindrical 
surface rotating with angular velocity a> about an axis parallel to its length. 

Taking the origin in the axis of rotation, and the axes of a;, y in a per- 
pendicular plane, then, with the same notation as before, d^jds will be equal 
to the normal component of the velocity of the boundary, or 

dJt dr 

if r denote the radius vector from the origin. Integrating we have, at all 
points of the boundary, 

i/f = ^r* -f const (1) 

If we assume any possible form of ^, this will give us the equation of a 
series of curves, each of which would, by rotating round the origin, produce 
the system of stream-lines determined by tf/. 

As examples we may take the following : 

r. If we assume ^= Ar^ coa2e = A {x^ - y% (2) 

the equation (1) becomes 

(icu - ^) x2 -f (icu -f A) y^ = C, 

which, for any given value of 4, represents a system of similar conies. That 
this system may include the elUpse 

* Prof. Hole Shaw has made a number of beautiful experimental delineations of the forms 
of the stream -lines in cases of steady irrotational motion in two dimensions, including those 
figured on pp. 76, 81; see Trans, Inat, Nav. Arch. t. xl. (1898). The theory of his method 
will find a place in Chapter xi. 

t When the general direction of the stream makeB an angle a with the lamina the couple is 
i^P9o*c* sin 2a. Cisotti, Ann. di mat. (3), t. xix. p. 83 (1912). 



71-72] Botating Bomidary 83 

we must have ( Ja> — A)a*= (J<«i + A) 6*, 

Hence the formula ^ = Jcu . - ^ ,^ (x* — y*) . . . 5r (3) 

gives the motion of a liquid contained within a hollow cylinder whose section 
is an ellipse with semi-axes a, 6, produced by the rotation of the cylinder 
about its longitudinal axis with angular velocity co. The arrangement of 
the stream-lines ^ = const, is shewn in the figure on p. 85. 

The corresponding formula for <f> is 

a* - 6« 

^^-^-^njTp-^ W 

The kinetic energy of the fluid, per unit length of the cylinder, is given by 

^^ - "//Kir - 01 "»=» ^^■"— '«'•••• ■<») 

This is less than if the fluid were to rotate with the boundary, as one rigid 
mass, in the ratio of 



W + bV 



to unity. We have here an illustration of Lord Kelvin's minimum theorem, 
proved in Art. 45.- 

2^. Let us assume 

^ = 4r» cos 3d = -4 (x» - 3ay*). 

The equation (1) of the boundary then becomes 

^ (x» - 3xy«) - icu (a;« + y2) = C (6) 

We may choose the constants so that the straight hne x=^ a shall form part 
of the boundary. The conditions for this are 

Aa^ - iwa^ = C, ^Aa 4- Jo) = 0. 

Substituting the values of Ay C hence derived in (6), we have 

a^ - a» - ^xy^ -h 3a (x^ - a« + y^) = 0. 

Dividing out by x — a, we get 

x^+iax^- 4a« = 3y«, 
or a; 4- 2a = ± \/3 • V- 

The rest of the boundary consists therefore of two straight lines passing 
through the point (— 2a, 0), and inclined at angles of 30° to the axis of x. 

We have thus obtained the formulae for the motion of the fluid contained 
within a vessel in the form of an equilateral prism, when the latter is rotating 

6—2 



84 Motion of a Liquid in Two Dimensions [chap, iv 

with angular velocity w about an axis parallel to its length and passing 
through the centre of its section; viz. we have 

V'^-i^^cosSi?, <^ = ^~r»sin3d, (7) 

where 2 \/3a is the length of a side of the prism*. 

3°. In the case of a liquid contained in a rotating cylinder whose section 
is a circular sector of radius a and angle 2a, the axis of rotation passing 
through the centre, we may assume 

, „ cos 2d ^^ /|.\(2n+l)ir/2« -g 

the middle radius being taken as initial line. For this makes ^ = |a>r^ for 
d = ± a, and the constants -^jn+i can be determined by Fourier's method so as 
to make ^ = ^wd^ for r = a. We find 

^«n+i = (-)»+^ oM* l(2n+l)^-4a - (2n+l)fl- ^ (2n+l)^ + 4a} ' * " <^^ 
The conjugate expression for ^ is 

The kinetic energy is given by 

2T=~p|<^^<fe=~ 2pcjj\rdr, (11) 

where <f>a denotes the value of <f>ioT = a, the value of d<f>/dn being zero over 
the circular part of the boundary f. 

The case of the semicircle a = Jtt will be of use to us later. We 

then have 

a>a2 f 1 2 1 



^2n+l — {"^r^ - 1o^ 1 " O*, 1 1 "^ 



TT I2n - 1 2n 4- 1 2n + 3 
and therefore 



(12) 



J ^-^^'" ^ T" ^ 2;r+3 12;^^^: ■" 2im "^ 2;rT3! ^ " "^ l^ " s" J • 



Hence J 27 = J7r/)a>«a* (^^ " i) = 'SlOGa^ x J7r/)a>^2 (13) 

This is less than if the fluid were solidified, in the ratio of *6212 to 1. See 
Art. 45. 

* The problem of fluid motion in a rotating cylindrical case is to a certain extent mathe- 
matically identical with that of the torsion of a uniform rod or bar. The above examples are 
mere adaptations of two of de Saint- Venant*8 solutions of the latter problem. See Thomson and 
Tait, Art. 704 et seq, 

t This problem was first solved by Stokes, " On the Critical Values of the Sums of Periodic 
Series," Camb. Trane. t. viii. (1847) [Papers, t. i. p. 306]. See also Hicks, Jfe**. of Math, t. viii. 
p. 42 (1878); Greenhill, (bid. t. viii. p. 89, and t. x. p. 83. 

X Greenhill, l.c. 



72] 



Rotating Cylinder 



85 



4°. With the same notation of elliptic co-ordinates as in Art. 71, 3°, let 

us assume 

^ + i^ = (7i6-2(^+*^> (14) 

Since x* + y* = \c^ (cosh 2^ + cos 217), 

the equation (1) becomes 

Ce"*^ cos 2?^ — Jcuc^ (cosh 2^ 4- cos 2iy) *= const. 
This system of curves includes the ellipse whose parameter is |^o> provided 

Cc-afo ^ jeuc* = 0, 
or, using the values of a, 6 already given, 

C = icu (a + h)\ 

80 that ^ = i<o (a + 6)^ e"^ cos 2?^, ' 

^ = icu (a + 6)^ c-2f sin 2^7. j 

At a great distance from the origin the velocity is of the order l/r*. 

The. above formulae therefore give the motion of an infinite mass of liquid, 
otherwise at rest, produced by the rotation of an elliptic cylinder about its 
axis with angular velocity co*. The diagram shews the stream-lines both 
inside and outside a rigid eUiptical cylindrical case rotating about its axis. 



(15) 




The kinetic energy of the external fluid is given by 

2T = ^pc* . a>2 (16) 

It is remarkable that this is the same for all confocal elliptic forms of the 
section of the cylinder. 

* Qiutri, Joum. Math. t. xiv. (1875); see also Beltrami, Lc, ante p. 81. 



86 Motion of a Liquid in Two Dimefnsions [chap, iv 

Combining these results with those of Arts. 66, 71 we find that if an 
elliptic cylinder be moving with velocities J7, V parallel to the principal axes 
of its cross-section, and rotating with angular velocity a>, and if (further) the 
fluid be circulating irrotationally round it, the cyclic constant being ic, then 
the stream-function relative to the aforesaid axes is 

The foihs followed by the particles of fluid, as distinguished from the 
stream-lines, in several of the preceding cases, have been studied by Prof. 
W. B. Morton f; they are very remarkable. The particular case of the 
circular cylinder (Art. 68) was examined by Maxwell]:. 

Steady Motions vnth a Free Surface. 

73. The first solution of a problem of two-dimensional motion in which 
the fluid is bounded partly by fixed plane walls, and partly by surfaces 
of constant pressure, was given by Helmholtz§. KirchhoS|| and others 
have since elaborated a general method of dealing with such questions. If 
the surfaces of constant pressure be regarded as free, we have a theory of 
jets, which furnishes some interesting results in illustration of Art. 24. 
Again, since the space beyond these surfaces may be filled with liquid at 
rest, without altering the conditions of the problem, we obtain also a number 
of cases of 'discontinuous motion,' which are mathematically possible with 
perfect fluids, but whose practical significance is less easily estimated. We 
shall return to this point at a later stage (Chap, xi) ; in the meantime we 
shall speak of the surfaces of constant pressure as 'free.' Extraneous forces, 
such as gravity, being neglected, the velocity must be constant along any 
such surface, by Art. 21 (2). 

The method in question is based on the properties of the function C 
introduced in Art. 65. The moving fluid is supposed bounded by stream- 
lines iff = const., which consist partly of straight walls, and partly of lines 
along which the resultant velocity (q) is constant. For convenience, we may 
in the first instance suppose the units of length and time to be so adjusted 
that this constant velocity is equal to unity. Then in the plane of the 
function f the lines for which q = 1 are represented by arcs of a circle of unit 
radius, having the origin as centre, and the straight walls (since the direction 

* The case of a oylindrical lamina whose Bection is an arc of a circle, with circulation round it 
is solved by Kutta, 8itzb, d. k. bayr. Akad, d. Wiss. 1910; some related problems are discussed 
by Blasius, ZeiUchr.f, Math, u. Phya. t. liz. p. 226 (1911). 

t Proc. Roy. 8oc. A. t. Ixzxix. p. 106 (1913). 

t Proc. Lond. Math. 8oc. t. iii. p. 82 (1870) [Papers, t. ii. p. 208]. 

§ Loc. ciL ante p. 21. 

I) "Zur Theorie freier Flussigkeitsstrahlen," CreUe, t. Ixz. (1869) [Gea. Abh. p. 416]. See also 
his Mechanik, co. xzi., zxii. 



72-73] Free Stream-Lines 87 

of the flow along each is constant) by radial lines drawn outwards from the 
circumference. The points where these lines meet the circle correspond to 
the points where the bounding stream-lines change their character. 

Consider, next, the function log ^. In the plane of this function the 
circular arcs for which j = 1 become transformed into portions of the 
imaginary axis, and the radial lines into lines parallel to the real axis, since 
if { = q-^ e^ we have 

logC = log^ + ie (1) 

It remains, then, to determine a relation of the form* 

logC=fM, (2) 

where w = <f> -\- up, bs usual, such that the rectilinear boundaries in the plane 
of log ^ shall correspond to straight lines iff = const, in the plane of w. 
There are further conditions of correspondence between special points, one 
on the boundary, and one in the interior, of each region, which render the 
problem determinate. 

When the correspondence between the planes of ^ and w has been 
estabUshed, the connection between z and t<7 is to be found, by integration, 
from the relation 

£--«• o 

The arbitrary constant which appears in the result is due to the arbitrary 
position of the origin in the plane of z. 

The problem is thus reduced to one of conformal representation between 
two areas bounded by straight lines f. This is resolved by the method of 
Schwarz and ChristofEel, already referred to J, in which each area is repre- 
sented in turn on a half-plane. Let Z {= X -^ iY) and t be two complex 
variables connected by the relation 

^ = 4 (a - t)-^l- {b - t)-^i^ (c - t)-fi^. . . , (4) 

where a,b, c, ... are real quantities in ascending order of magnitude, whilst 
a, ]3, y, ... are angles (not necessarily all positive) such that 

a + P + y^ ... =27r; (5) 

and consider the line made up of portions of the real axis of t with small 

semi-circular indentations (on the upper side) about the points a, b, c, 

If a point describe this line from t = — oo toi= + oo, the modulus only 
of the expression in (4) will vary so long as a straight portion is being 

* The use of log i*, in place of tt is dne to Planck, Wied. Ann. t. xxi. (1884). 
t See Forsyth, Theory of Functions, c. xx, 
{ See the second footnote on p. 78 ante. 



88 Motion of a Liquid in Two Dimensions [chap, iv 

described, whilst the e£Eect of the clockwise description of the semi-circular 
portions is to introduce factors 6**, e*^, e% ... in succession. Hence, regarding 
dZ/dt as an operator which converts 8^ into 8Z, we see that the upper half 
of the plane of t is conformably represented on the area of a closed polygon 
whose exterior angles are a, j8, y, . . . , by the formula 

Z = Al(a - t)-^i^ (b-t)-^'^ (c - t)-^l^ ...dt + B, (6) 

provided the path of integration in the ^plane lies wholly within the region 
above delimited. When a, by c, . . , , a, j8, y, ... are given, the polygon is 
completely determinate as to shape ; the complex constants A, B only affect 
its scale and orientation, and its position, respectively. 

As already indicated, we are specially concerned with the conformal 

representation of rectangular areas. If a = j8 = y = 8 = j7r, the formula (6) 

becomes 

r dj 

^ = ^ J V{(a - t) {b -t)(c- t) (d -t)}'^^ ^^^ 

It is easily seen that the rectangle is finite in all its dimensions unless two at 
least of the points a, 6, c, d are at infinity. The excepted case is the one 
specially important to us; the two finite points may then conveniently be 
taken to be t = ± 1, so that 

= A cosh-i t + B (8) 

In particular, the assumption ^ 

t = cosh jy (9) 

where k is real, transforms the space bounded by the positive halves of the 
lines Y = 0, Y = irk, and the intervening portion of the axis of Y, into the 
upper half of the plane t, Cf. Art. 66, 1°. 

Again, if the two finite points coincide, say at the origin of t, we have 



Z = AJj + B = Alogt-hB (10) 



This transforms the upper half of the t-plane into a strip bounded by two 
parallel straight hues. For example, if 

« = e^/*, (11) 

where k is real, these may be the lines Y = 0, Y = irk, 

74. As a first appUcation of the method in question, we may take the 
case of a fluid escaping from a large vessel by a straight canal projecting 
inwards*. This is the two-dimensional form of Borda's mouthpiece, referred 
to in Art. 24. 

* This problem was first solved by Helmholtz, Lc. ante p. 21. 



73-74] Borda's Mouthpiece 89 

The boundaries of corresponding areas in the planes of ^, log ^, and w, 
respectively, are easily traced, and are shewn in the figures'*'. It remains to 
connect the areas in the planes of log ^ and w each with the upper half- 
plane of an intermediate variable L It appears from equations (8) and (10) 
of the preceding Art. that this is accomplished by the substitutions 

logC^ A cosh-i t + B, w = Clogt + D (1) 

We have here made the comers A, A' in the plane of log ^ correspond to 
t = ±ly and we have also assumed that f = corresponds to i/? = — oo , as is 
evident on inspection of the figures. To specify more precisely the values of 
the cyclic functions cosh~^ t and log t we will assume that they both vanish 
at ^ =: 1, and that their values at other points in the positive half -plane are 



A 



-r' 



/ 



^ 







I 
I 

■J' I 



i t 



I 



2 B' 



W 
A 



J. ^' 



A' B' 

determined by considerations of continuity. It follows that when < = — 1 the 
value of each function will be vrr. At the points A', A in the plane of log ^, 
we have, on the simplest convention, log ^ = and 2tTr, respectively ; whence, 
towards determining the constants in (1) we have 

= 5, 2i7T = vttA -h B, 

so that log C = 2 cosh-i t (2) 

Again, in the plane of w we take the line //' as the line ^ = 0; and if the 
final breadth of the issuing jet be 26, the bounding stream-lines will be 
iff = ±b. We may further suppose that ^ = is the equipotential curve 
passing through A and A\ Hence, from (1) 

= ittC + D, -ib^D, 



80 that w = — \ogt — ib (3) 

* The heavy lines oonespond to rigid boundaries, and the fine continuous lines to free 
surfaces. Corresponding points io the various figures are indicated by the same letters. 



90 Motion of a Liquid in Two Dimensions [chap, iv 

It is easy to eliminate t between (2) and (3), and thence to find the relation 
between z and w by integration, but the formulae are perhaps more 
convenient in their present shape. 

The course of either free stream-line, say A'ly from its origin at A\ is now 
easily traced. For points of this Une t is real, and ranges from 1 to 0; we 
have, moreover, from (2), iff = 2 cosh"* «, or < = cos \6, Hence, also, from (3), 

^ = — log cos \9 (4) 

Since, along this line, we have d<f>ld8 = — j = — 1, we may put^ = — «, where 
the arc 8 is measured from A\ The intrinsic equation of the curve is 
therefore 

5 = — log sec i& (5) 

IT 

From this we deduce in the ordinary way 

x = -(8in2J0-logsecje), y = -(0-sin0), (6) 

TT TT 

if the origin be at A', By giving 6 a series of values ranging from to tt. 
the curve is easily plotted. 



Line of Symmetry. 

Since the asymptotic value of y is 5, it appears that the distance between 
the fixed walls is 46. The coefficient of contraction is therefore J, in accord- 
ance with Borda's theory. 

75. The solution for the case of fluid issuing from a large vessel by an 
aperture in a plane wall is analytically very similar. The chief difference is 
that the values of log ^ at the points A^ A' in the figures must now be taken 
to be and — iir, respectively, whence, to determine the constants -4, B 
in (1) we have 

so that log 5 = cosh-^ t — vrr (7) 



74-761 



Vena Contracta 



91 



The relation between w and t is exactly as before, viz. 

t£7 = — log < — io, 

IT 



(8) 



where 26 is the final breadth of the stream, between the free boundaries. 



z 



r 

I 



A' 



M r^ 



hgt 



B' 



-r 



B' 



W 
A 




-/' 



For the stream-line -4/, t is real, and ranges from — 1 to 0. Since, also, 
i6 = cosh"^ ^ — ijT we may put t = cos (d 4- ^), where d varies from to — Jtt . 
Hence, from (8), with^ = — *, we have, for the intrinsic equation of the stream- 
line, 

« = ?^ log (- sec e) (9) 



TT 



Line of Symmetry. 



92 



Motion of a Liquid in Two Dimensions [chap, iv 



From this we find 

46 



26 



x = — sin* ^0, y = — {log tan (\tt + \0) — sin 0), 

TT IT 



(10) 



if the point A in the first diagram be taken as origin'*'. The curve is shewn 
(in an altered position) at the foot of the preceding page. 

The asymptotic value of a?, corresponding to = — Jtt, is 26/7r, the half 
width of the aperture is therefore {it + 2) 5/Tr, and the coefficient of con- 
traction is 

7r/(7r + 2) = -611. 

76. In the next example a stream of infinite breadth is supposed to 
impinge directly on a fixed plane lamina, and thence to divide into two 
portions bounded internally by free surfaces. 

The middle stream-Une, after meeting the lamina at right angles, branches 
off into two parts, which follow the lamina to the edges, and thence form the 
free boundaries. Let this be the line = 0, and let us further suppose that 
at the point of divergence we have ^ = 0. The forms of the boundaries in 
the various planes are shewn in the figures. The region occupied by the 



X C! A 





logl 
A c 



A* 




A 
A ^ 



/ A' 
— 1 



3_ 



moving fluid now corresponds to the whole of the plane w, which must 
however be regarded as bounded internally by the two sides of the line 
= 0, <^ < 0. 

With the same conventions as in the beginning of Art. 75, we have 

log 5 = cosh"^ t — im, (1) 



or 



« = -cosh(logO = -j(C+^). 



(2) 



* This example was given by Kirchhoff (Z.c.)» and discussed more fully by Rayloigh, "Notes 
on Hydrodynamics,*' PhiL Mag. Dec. 1876 [Papers, t. i. p. 297]. 



75-76] Impact of a Stream on a Lamina 93 

The correspondence between the planes of w and t is best established by 
considering first the boundary in the plane of w~^. The method of Schwarz 
and ChristofEel is then at once applicable. Putting a = — tt, j8 = y= ... =0, 
in Art. 73 (4), we have 



IT 



= At, w-i = J4^* + B. 



(3) 



At I we have < = 0, vr^ — 0, so that S = 0, or (say) 

C 



«;=--,. 



(4) 



To connect C (which is easily seen to be real) with the breadth (i) of the 
lamina, we notice that along CA we have { == g~^, and therefore, from (2) 

«=-i(^ + ?)> g=-<-\/(^«-l), (5) 

the sign of the radical being determined so as to make 9 = f or ^ = — oo . 
Also, dxld<f> = — Ijq. Hence, integrating along CA in the first figure 
we have 



I 



J -00 



^f*-*«/: 



-1 dt 



qt* 



00 



= -4C|' J- < + V(<* - 1)}*, • .(6) 



whence 



I 






(7) 



Line of Symmetry. 



Along the free boundary AI, we have log { = i^, and therefore, from (2) 
and (4), 

t^-cwd, <I> = -Caw*e (8) 



94 



Motion of a Liquid in Two Dimensions [chap, iv 



The intrinsic equation of the curve is therefore 

8 = — —7 sec^ 6. 

TT + 4 

where 9 ranges from to — \tt. This leads to 

21 
a? = — , -7 (sec e + i^r), 



(9) 



y = 



I 



7r + 4 



{sec tan - log (iw + \e)), 



(10) 



the origin being at the centre of the lamina. The curve is shewn on the 
preceding page. 

The excess of pressure on the anterior face of the lamina is, by Art. 23 
(7), equal to Jp (1 — g*). Hence the resultant force on the lamina is 



'>/;l<'-'^S*--W!l(r')?--W->"'-"?-'"*- 

(11) 

It is evident from Art. 23 (7), and from the obvious geometrical similarity 
of the motion in all cases, that the resultant pressure (Pq? ^^J) ^U vary as 
the square of the general velocity of the stream. We thus find, for an 
arbitrary velocity jo*> 

— V~4. P?o* • ^ = '440/)go* A (12) 



P« = 



77. If the stream be oblique to the lamina, making an angle a, say, with 
its plane, the problem is modified in the manner shewn in the figures. 



/y 



I ■ 



/ 

/ 



C^A 




logl 



o 



, ? 




7 A' 



W 
A' 



W 



— 1 



> 



• Kirchhoff, l,c. ante p. 86; Rayleigh, "On the Resistance of Fluids," Phil Mag. Dec. 1876 
[Papers, 1. 1. p. 287]. 



76-77] Pressure on a Lamina 95 

The equations (1) and (2) of the preceding Art. still apply; but at the point / we now 
have f =«~* , and therefore t = cos a. Hence, in place of (4) *, 

«^=-7i— — ,. (13) 

(t - cos of 

At points on the front face of the lamina, we have, since q-'^=\(\, 

^= ±f+V(^«-l), g= ±<"V(«*-1) (14) 

where the upper or the lower signs are to be taken, according as < ^ 0, i.e, according as the 
point referred to lies to the left or right of C in the first figure. Hence 

Between A' and C, t varies from 1 to oo, whilst between A and C the range is from 
- 00 to - 1. If we put 

1 - cos a cos a> 
cos a — cos a> 

the corresponding ranges of » will be from ir to a, and from a to 0, respectively ; and we 
find 

dt cos a - COB « . , . ,/^« 1 V sin a sin ft) 

rj r»= r-7 SlU ftlOft), ± ^(i^-l) = . 

(t - COS ay sm* a ^ x / ^^g ^ _ ^^^ ^ 

Hence j-= - . i (1 - cos a cos m + sin a sin <») sin a>, (16) 

and therefore 

a; = . . {2 cos «> +cos a sin* tt> +sin a sin » cos ta+ihr - «) sin a}, (17) 

where the origin has been adjusted so that x shall have equal and opposite values when 
«» =0 and « =7r, respectively ; t.e it has been taken at the centre of the lamina. Hence, in 
terms of C, the whole breadth is 

/=*+'^.C. (18) 

The distance, from the centre, of the point (o =a) at which the stream divides is 

_ 2cosa(l +sin*a)+(^ir -a) sin a . 
~ 4+irCOSa 

To find the total pressure on the front face, we have 



4+9rC06a 



= r-5~ . Sm" ttCfo). 



• 3 .™ w.-, (20) 

sm' o ^ ' 

Integrated between the h'mits ir and 0, this gives TpC/sin' a. Hence, in terms of I, and of 
an arbitrary velocity gg of the stream, we find 

^ ^^rina_ ^ (21) 

* The solution up to this point was given by Kirchhoff {Crelle, Lc); the subsequent discus- 
sion is taken, with merely analytical modifications, from the paper by Rayleigh. 



96 Motion of a Liquid in Two Dimensions [chap, rv 

To find the centre of pressure, we take moments about the centre of the lamina. Thus 

\p 1(1 -(f')xdx= - .^ . / .r8in'a>(2a> 
3 8m» a y » 

^ irpO Ccosa 

sin' a * 8in*a ' ^ ' 

on substituting the value of x from (17). The first factor represents the total pressure; 
the abscissa x of the centre of pressure is therefore given by the second, or, in terms of 
the breadth 



. cos a , 
x-\- — .— ./. 
4 +7rsma 



(23) 



In the following table, derived from Rayleigh*s paper, the column I gives the excess 
of pressure on the anterior face, in terms of its value when a =00°; whilst columns II and 
III give respectively the distances of the centre of pressure, and of the point where the 
stream divides, from the centre of the lamina, expressed aa fractions of the total breadth *. 



a 


I 


II 


ni 


90° 


1000 


•000 


•000 


70° 


•965 


•037 


•232 


60° 


•854 


•075 


•402 


30° 


•641 


•117 


•483 


20° 


•481 


•139 


•496 


10° 


•273 


•163 


•500 



78. An interesting variation of the problem of Art. 76 has been discussed 
by Bobylefif. A stream is supposed to impinge symmetrically on a bent 
lamina whose section consists of two equal straight lines forming an angle. 

If 2a be the angle, measured on the down-stream side, the boundaries in the plane of ^ 
can be transformed, so as to have the same shape as in the Art. cited, by the assumption 

provided A and n be determined so as to make f = 1 when f =e"'****~*^ and f sc"*"" when 
f =€-«*»+•). This gives 

On the right-hand half of the lamina, i will be negative as before, and since 9~^ = | CL 






(24) 



Henoe 






di 



<V(«*-1)' 
rf^ 

iV(««-l)* 



* For the compaiison with oxporimental results see Rayleigh, Z.r. and Nature, t. xlv. (1891) 
[Papers, t. iii. p. 491]. 

t Journal of the Ruseian Physico-Chemuxd Socieiy, t. xiii. (1881) [Wiedemann's BeibldUcr, 
t. vi. p. 163]. The problem appears, however, to have been previously discussed ia a similar 
manner by )L R^thy, Klausenburger Berichte, 1879. It is generalized by Bryan and Jones, 
Proc, Boy, Soe. A, t. xci. p. 354 (1915). 



77-78] Bobykff's Problem 97 

These can be reduced to known forms by the substitution 
where » ranges from to 1. We thus find 






(26) 



(26) 



We have here used the formulae 



yo(i+«)* yoi + « 

/ 71— — r=a« =-*+«/ = a«j>, 

jo(i+«)* ' yoi+» 

where 1 > ifc > 0. 

Since, along the stream-line, d^ldtf) = - l/g, we have from (25), if 6 denote the half- 
breadth of the lamina. 



. ^f, 2a 4a« P4>^'' , 1 



(27) 



The definite integral which occurs in this expression can be calculated from the formula 

!fH'^- ii-t)'(8-t) ^**"-w-i*«-<') m 



/ 



where "^ (m), =d/dfn . log n (m), is the function introduced and tabulated by Gauss*. 
The normal pressure on either half is, by the method of Art. 76, 



^'^ J ^oo \q J at " '^jol+w '^sm Jwir '^ ir sm a 

The resultant pressure in the direction of the stream is therefore 

4a* 



(29) 



ir 



pC, (30) 



Hence, for any arbitrary velocity Qq of the stream, the rteultant pressure is 

P^^.pqo^b (31) 

where L stands for the numerical factor in (27). 

For a =in, we have L =2 + jfr, leading to the same result as in Art. 76 (12). 

In the following table, taken (with a slight modification) from Bobyleff's paper, the 
second column gives the ratio P/Pq of the resultant pressure to that experienced by a 
plane strip of the same area. This ratio is a maximum when a = 100°, about, the lamina 
being then concave on the up-stream side. In the third column the ratio of P to the dis- 
tance (26 sin a) between the edges of the lamina is compared with ipQoK For values of a 
nearly equal to 180°, this ratio tends to the value unity, as we should expect, since the fluid 
within the acute angle is then nearly at reflt, and the pressure-excess therefore practically 

* '* DisqaisitioQos generalos circa seriem infinitam ...,"' Wtrire, Qottingen, 1S70 . . . , t. iii. p. 161. 
L. H. 7 



98 



Motion of a Liquid in Two Dimensions [chap, iv 



equal to ipqo** The last column gives the ratio of the resultant pressure to that experi- 
enced by a plane strip of breadth 26 sin a, as calculated from (12). 



a 


PIP, 


Plpq^h sin a 


P/Po8ina 


10° 


•030 


•199 


•227 


20° 


•140 


•359 


•409 


30° 


•278 


•489 


•555 


40° 


•433 


•593 


•674 


45° 


•612 


•637 


•724 


50° 


•589 


•677 


•769 


60° 


•733 


•745 


•846 


70° 


•854 


•800 


•909 


80° 


•945 


•844 


•959 


90° 


1^000 


•879 


1000 


100° 


1016 


•907 


1031 


110° 


•995 


•931 


1059 


120° 


•935 


•950 


1079 


130° 


•840 


•964 


1096 


135° 


•780 


•970 


1103 


140° 


•713 


•975 


1109 


150° 


•559 


•984 


1119 


160° 


•385 


•990 


1126 


170° 

1 


•197 


•996 


1132 



Discontinuous Motions, 

79. It must suffice to have given a few of the more important examples 
of steady motion with a free surface, treated by what is perhaps the most 
systematic method. Considerable additions to the subject have been made 
by Michell*, Lovef, and other writers J. It remains to say something of the 
physical considerations which led in the first instance to the investigation of 
such problems. 

We have, in the preceding pages, had several instances of the flow of 
a Uquid round a sharp projecting edge, and it appeared in each case that the 
velocity there was infinite. This is indeed a necessary consequence of the 
assumed irrotational character of the motion, whether the fluid be incom- 
pressible or not, as may be seen by considering the configuration of the 

« "On the Theory of Free Stream-lines/' Phil, Trans, k, t. clxxxi. (1890). 

t '*0n the Theory of Discontinuous Fluid Motions in Two Dimensions," Proc, Camb, Phil, 
Soc, t, viL (1891). 

X For references see Love, Encycl. d. maih, Wi«s. t. iv. (3), pp. 97 ... . A very complete 
aoconnt of the more important known solutions, with fresh additions and developments, is given 
by GreenhiU, Report on the Theory of a Stream-line past a Plane Barrier, published by the Advisory 
Committee for Aeronautics, 1910. 

The extension to the case of curved rigid boundaries is discussed in a general manner in various 
papers by Levi-Civita and OisottL For these, reference may be made to the Rend, d, Circolo 
Mat, di Palermo, tt, xxiii. xxv. xxvi. xxviii and the Rend, d. r, Accad, d, Lincei, tt. xx. xxi. ; 
the working out of particular cases naturally presents great difficulties. The theory of mutually 
impinging jets is treated very fully by Gisotti, " Vene confluenti," Ann, di mat, (3), t. xxiii. 
p. 285 (1914). 



78-79] Discontinuous Motions 99 

equipotential surfaces (which meet the boundary at right angles) in the 
immediate neighbourhood. 

The occurrence of infinite values of the velocity may be avoided by 
supposing the edge to be slightly rounded, but even then the velocity near 
the edge will much exceed that which obtains at a distance great in 
comparison with the radius of curvature. 

In order that the motion of a fluid may conform to such conditions, it is 
necessary that the pressure at a distance should greatly exceed that at the 
edge. This excess of pressure is demanded by the inertia of the fluid, which 
cannot be guided round a sharp curve, in opposition to centrifugal force, 
except by a distribution of pressure increasing with a very rapid gradient 
outwards. 

Hence, unless the pressure at a distance be very great, the maintenance of 
the motion in question would require a negative pressure at the comer, such 
as fluids under ordinary conditions are unable to sustain. 




To put the matter in as definite a form as possible, let us imagine the 

following case. Let us suppose that a straight tube, whose length is large 

compared with the diameter, is fixed in the middle of a large closed vessel 

filled with frictionless liquid, and that this tube contains, at a distance from 

the ends, a sUding plug, or piston, P, which can be moved in any required 

manner by extraneous forces appUed to it. The thickness of the walls of the 

tube is supposed to be small in comparison with the diameter ; and the edges, 

at the two ends, to be rounded off, so that there are no sharp angles. Let us 

further suppose that at some point of the walls of the vessel there is a lateral 

tube, with a piston Q, by means of which the pressure in the interior can be 

adjusted at will. 

7—2 



100 Motion of a Liquid in Two Dimensions [chap, iv 

Everything being at rest to begin with, let a slowly increasing velocity be 
communicated to the plug P, so that (for simplicity) the motion at any 
instant may be regarded as approximately steady. At first, provided a 
sufficient force be applied to Q, a continuous motion of the kind indicated in 
the diagram on p. 72 will be produced in the fluid, there being in fact only 
one type of motion consistent with the conditions of the question. As the 
acceleration of the piston P proceeds, the pressure on Q may become 
enormous, even with very moderate velocities of P, and if Q be allowed to 
yield, an annidar cavity will be formed at each end of the tube. 

It is not easy to make out the further course of the motion in such a case 
from a theoretical standpoint, even in the case of a 'perfect' fluid. In actual 
liquids the problem is modified by viscosity, which prevents any slipping of 
the fluid immediately in contact with the tube, and must further exercise 
a considerable influence on such rapid differential motions of the fluid as are 
here in question. 

As a matter of observation, the motions of fluids are often found to 
differ widely, under the circumstances supposed in each case, from the types 
represented in such diagrams as those of pp. 71, 72, 80, 81. In such a case 
as we have just described, the fluid issuing from the mouth of the tube does 
not immediately spread out in all directions, but forms, at all events for some 
distance, a more or less compact stream, bounded on all sides by fluid nearly 
at rest. A familiar instance is the smoke-laden stream of gas issuing from a 
chinmey. In all such cases, however, the motion in the immediate neighbour- 
hood of the boundary of the stream is found to be wildly irregular*. 

It was the endeavour to construct types of steady motion of a frictionless 
liquid, in two dimensions, which should resemble more closely what is 
observed in such cases as we have referred to, that led Helmholtzf and 
Kirchhoff t to investigate the theory of free stream-lines. It is obvious that 
we may imagine the space beyond a free boundary to be occupied, if we 
choose, by liquid of the same density at rest, since the condition of constant 
pressure along the stream-line is not thereby affected. In this way the 
problems of Arts. 76, 77, for example, give us a theory of the pressure 
exerted on a fixed lamina by a stream flowing past it, or (what comes to the 
same thing) the resistance experienced by a lamina when made to move with 
constant velocity through a liquid which would otherwise be at rest. 

The question as to the practical validity of this theory will be referred to 
later in connection with some related problems (Chapter xi.). 

* Recent experiments would indicate that jets may be formed htfore the limiting velocity of 
Helmholtz ia reached, and that viscosity plays an essential part in the process. Smoluchowski. 
**Sur la formation des veines d'efflux dans les liquides/' BuU. de VAcad, de Cracovie, 1904. 

t U. c. ante pp. 21, 86. 



79-80] Flow in a Curved Stratum 101 

Fhw in a Curved Stratum, 

80. The theory developed in Arts. 59, 60 may be readily extended to 
the two-dimensional motion of a curved stratum of liquid, whose thickness is 
small compared with the radii of curvature. This question has been discussed, 
from the point of view of electric conduction, by Boltzmann*, Kirchhofff* 
ToplerJ, and others. 

As in Art. 59, we take a fixed point A, and a variable point P, on the 
surface defining the form of the stratum, and denote by \fs the flux across any 
curve AP drawn on this surface. Then ^ is a function of the position of P, 
and by displacing P in any direction through a small distance Ss, we find that 
the flux across the element 8^ is given by d^jds . hs. The velocity perpen- 
dicular to this element will be hfsjhhsy where h is the thickness of the 
stratum, not assumed as yet to be imiform. 

If, further, the motion be irrotational, we shall have in addition a velocity- 
potential <f>, and the equipotential curves ^ = const, will cut the stream-lines 
= const, at right angles. 

In the case of uniform thickness, to which we now proceed, it is convenient 
to write ^ for ^/A, so that the velocity perpendicular to an element hs is now 
given indifferently by dilsjds and d<f>idn, Sn being an element drawn at right 
angles to hs in the proper direction. The further relations are then exactly as 
in the plane problem ; in particular the curves <f> = const., ^ = const., drawn 
for a series of values in arithmetic progression, the common difference being 
infinitely small and the same in each case, will divide the surface into 
elementary squares. For, by the orthogonal property, the elementary spaces 
in question are rectangles, and if Ssi, Ss^ be elements of a stream-fine and 
an equipotential fine, respectively, forming the sides of one of these 
rectangles, we have 3^/3*2 = 3^/9*i> whence 8«i = 8«2, since by construction 
80 = &^. 

Any problem of irrotational motion in a curved stratum (of uniform 
thickness) is therefore reduced by orthomorphic projection to the corre- 
sponding problem in piano. Thus for a spherical surface we may use, among 
an infinity of other methods, that of stereographic projection. As a simple 
example of this, we may take the case of a stratum of uniform depth covering 
the surface of a sphere with the exception of two circular islands (which may 
be of any size and in any relative position). It is evident that the only (two- 
dimensional) irrotational motion which can take place in the doubly-connected 
space occupied by the fluid is one in which the fluid circulates in opposite 

• Wiener Sitzunggberichte, t. lii. p. 214 (1866) [WissenschafUiche Abhandlungen, Leipzig, 1909, 
t. i p. 1]. 

t BerL Monai^er. July 19, 1876 [Oes, Abh. p. 66]. 
t Pogg^ Ann. t. clx. p. 376 (1877). 



102 Motion of a Liquid in Two Dimensions [chap, iv 

directions round the two islands, the cyclic constant being the same in each 
case. Since circles project into circles, the plane problem is that solved in 
Art. 64, 2°, viz. the stream-lines are a system of coaxal circles with real 
'limiting points' (^, jB, say), and the equipotential lines are the orthogonal 
system passing through A^ B. Eetuming to the sphere, it follows from well- 
known theorems of stereographic projection that the stream-lines (including 
the contours of the two islands) are the circles in which the surface is cut by 
a system of planes passing through a fixed line, viz. the intersection of the 
tangent planes at the points corresponding to A and B, whilst the equipotential 
lines are the circles in which the sphere is cut by planes passing through 
these points*. 

In any case of transformation by orthomorphic projection, whether the 
motion be irrotational or not, the velocity (d^/dn) is transformed in the 
inverse ratio of a linear element, and therefore the kinetic energies of the 
portions of the fluid occupying corresponding areas are equal (provided, of 
course, the density and the thickness be the same). In the same way the 
circulation {fdift/dn . ds) in any circuit is unaltered by projection. 

* ThiB example is given by Kirohhoff, in the eleotrical interpretation, the problem considered 
being the distribution of current in a uniform spherical conducting sheet, the electrodes being 
situate at any two points A,Boi the surface. 



CHAPTER V 

IRROTATIONAL MOTION OF A LIQUID: PROBLEMS IN 

THREE DIMENSIONS 

81. Of the methods available for obtaining solutions of the equation 

VV = (1) 

in three dimensions, the most important is that of Spherical Harmonics. 
This is especially suitable when the boundary conditions have relation to 
spherical or nearly spherical surfaces. ^ 

For a full account of this method we must refer to the special treatises*, 
but as the subject is very extensive, and has been treated from different 
points of view, it may be worth while to give a shght sketch, without formal 
proofs, or with mere indications of proofs, of such parts of it as are most 
important for our present purpose. 

It is easily seen that since the operator V^ is homogeneous with respect 
to X, y, z, the part of <f> which is of any specified algebraic degree must satisfy 
(1) separately. Any such homogeneous solution of (1) is called a * spherical 
solid harmonic' of the algebraic degree in question. If <^„ be a spherical 
solid harmonic of degree n, then if we write 

<l>n = r-S,. (2) 

S„ will be a function of the direction (only) in which the point (x, y, z) lies 
with respect to the origin ; in other words, a function of the position of the 
point in which the radius vector meets a imit sphere described with the origin 
as centre. It is therefore called a 'spherical surface harmonic' of order nf. 

♦ Todhunter, Functions oj Laplace, Lamij and Bessel, Cambridge, 1876. Ferrers, Spherical 
Harmonics, Cambridge, 1877. Heine, Handbuch der Kugdfiinciionen, 2nd ed., Berlin, 1878. 
Thomson and Tait, Nahtral Philosophy, 2nd ed., Cambridge, 1879, t. i. pp. 171-218. Byerly, 
Fourier*9 Series and Spherical, Cylindrical, and Ellipsoidal Harmonics, Boston, U.S.A. 1893. 
Whittaker, Modem Analysis, Cambridge, 1902. 

For the history of the subject see Todhonter, History of the Theories of Attraction, dec, 
Cambridge, 1873, t. 11. Also Wangerin, *'Theorie d. Kugelfmiktionen, n.s.w./* Encyd. d. 
math. Wiss. U ii. (1) (1904). 

t The symmetrical treatment of spherical solid harmonics in terms of Cartesian co-ordinates 
was introduced by Clebsoh, in a much neglected paper, Crelle, t. Izi. p. 195 (1863). It was 
adopted independently by Thomson and Tait as the basis of their exposition. 



104 Irrotational Motion of a Liquid [chap, v 

To any solid harmonic <^„ of degree n corresponds another of degree 
— w — 1, obtained by division by r*"+^ ; i.e. <f> = T'-^^-^<f>n is ako a solution of 
(1). Thus, corresponding to any spherical surface-harmonic S„, we have the 
two spherical solid harmonics r^S„ and r^"^iS„ . 

82. The most important case is when n is integral, and when the surface- 
harmonic 8^ is further restricted to be finite over the unit sphere. In the 
form in which the theory (for this case) is presented by Thomson and Tait, 
and by Maxwell*, the primary solution of (1) is 

<f>-i-Alr (3) 

This represents as we have seen (Art. 56) the velocity-potential due to 
a point-source at the origin. Since (1) is still satisfied when <f> is differ- 
entiated with respect to a?, y, or 2, we derive a solution 



^-« = ^(^ai + ^a^-^^3i)r (*) 



dy 

This is the velocity-potential of a double-source at the origin, having its axis 
in the direction (^ m, n); see Art. 56 (3). The process can be continued, 
and the general type of spherical solid harmonic obtainable in this way is 

3" 1 

hi ^8 9 ^8 being arbitrary direction-cosines. 

This may be regarded as the velocity-potential of a certain configuration 
of simple sources about the origin, the dimensions of this system being small 
compared with r. To construct this system we premise that from any given 
system of sources we may derive a system of higher order by first displacing 
it through a space ^hg in the direction (Z,, m^, w^), and then superposing the 
reversed system, supposed displaced from its original position through a space 
ih, in the opposite direction. Thus, beginning with the case of a simple 
source at the origin, a first appUcation of the above process gives us two 
sources 0+, 0_ equidistant from the origin, in opposite directions. The same 
process applied to the system 0+, 0_ gives us four sources 0++, O.^., 0+_, 

at the comers of a parallelogram. The next step gives us eight sources 

at the corners of a parallelepiped, and so on. The velocity-potential, at 
a great distance, due to an arrangement of 2^ sources obtained in this way, will 
be given by (5), if hrA = mfhyh^ . . , h„, m' being the strength of the original 
source at 0. The formula becomes exact, for all distances r, when 
hi, hz, . , . h„ are diminished, and m' increased, indefinitely, but so that 
A is finite. 

* ElectrieUy and Magnetism, c. iz. 



81-83] Spherical Harmonics 105 

The surface-harmonic corresponding to (5) is given by 

3» 1 
Sn = i4f"+^ - - -, (6) 

and the complementary solid harmonic by 

By the method of * inversion*,' applied to the above configuration of 
sources, it may be shewn that the solid harmonic (7) of positive degree n 
may be regarded as the velocity-potential due to a certain arrangement of 
2" simple sources at infinity. 

The lines drawn from the origin in the various directions (Z,, m,, w,) are 
called the * axes' of the solid harmonic (5) or (7), and the points in which 
these lines meet the unit sphere are called the ^poles' of the surface-harmonic 
Sn • The formula (5) involves 2n + 1 arbitrary constants, viz. the angular 
co-ordinates (two for each) of the n poles, and the factor A, It can be 
shewn that this expression is equivalent to the most general form of 
spherical surface-harmonic which is of integral order n and finite over the 
unit sphere f. 

83. In the original investigation of Laplace ;{:, the equation V^ = is 
first expressed in terms of spherical polar co-ordinates r, ^, co, where 

a: = r cos ^, y = r sin ^ cos a>, 2 = r sin d sin co. 

The simplest way of effecting the transformation is to apply the theorem of 
Art. 36 (2) to the surface of a volume-element rW . r sin ^8a> . 8r. Thus the 
difference of fiux across the two faces perpendicular to r is 

^ ( ^ . fW . r sin Qhii\ 8r. 
Similarly for the two faces perpendicular to the meridian (a> = const.) we find 

i(M.,8in^8a,.8r)sd, 
and for the two faces perpendicular to a parallel of latitude (5 = const.) 

5- ( — ' ^1^ . ^85 . 8r ) 8a>. 
Hence, by addition, 

This might of course have been derived from Art. 81 (1) by the usual method 
of change of independent variables. 

* Explained by Thomson and Tait, Natural PhikMophy, Art. 515. 

t Sylveeter, PhU, Mag, (5), t. ii. p. 291 (1876) [Mathematical Papers, Cambridge, 1904. .., 
t. iii. p. 37]. 

t "Th^rie de Tattraction des sph^roides et de la figure des plan^tes," MAn, de VAtad, toy. 
des Sciences, 11^ [Oeuvres Computes, Paris, 1878. . ., t. x. p. 341]; Micaniqve CAeOt, Livre 2«»*, 
c. it 



106 Irrotational Motion of a Liquid [chap, v 

If we now assume that <l> is homogeneous, of degree n, and put 

^'^'^'^ s-Ore(«^'^^t) + sT^^^" + «(" + l)^- = «' ••••(2) 

which is the general differential equation of spherical surface-harmonics. 
Since the product n (w + 1) is unchanged in value when we write — n — 1 for 
n, it appears that 

will also be a solution of (1), as already stated (Art. 81). 

84. In the case of symmetry about the axis of a:, the term d^SJdw^ 
disappears, and putting cos = fi we get 

^.{(i-'^*)'^}-'"^"'''^'^"^'' ^'^ 

the difierential equation of spherical 'zonal' harmonics*. This equation, 
containing only terms of two different dimensions in /x, is adapted for in- 
tegration by series. We thus obtain 

S -aU n (n + 1 ) (n - 2) w (n + 1) (w + 3) ) 

" " 1 1.2 ^ + 1.2.3.4 ^ ~ • • •) 

+ bL (n-l)(n + 2) (n-8)(n-l)(» + 2)(« + 4) 1 

+ ^ r ^ 1.2 :~z~ f" + 1.2.3.4.5 1^ "••■[• 

(2) 

The series which here present themselves are of the kind called 'hyper- 
geometric ' ; viz. if we write, after Gauss f, 



a.jS , a.a+l.j8.j8+l 



l.y 1.2.y.y-|-l 



X* 



^ a.a+l.a + 2.j8.j8+l./3+2 ^ , 

^ 1.2.3.y.y-|-l.y+2 "^ + ' ■ ■ ' 



(3) 



we have 

S„ = AF (- in, i + in,i, fx^) + BfiF (i - in, 1 + fw, f , ^«). . . (4) 

The series (3) is of course essentially coDvergent when x lies between and 1 ; but 
when ar = 1 it is convergent if, and only if, 

y-a-/3>0. 
In this case we have 

where n (m) is in Gauss's notation the equivalent of Euler's r (w + 1). 

* So called by Thomson and Tait, because the nodal lines {8^=(i) divide the unit sphere into 
parallel belts, 
t I'C, ante p. 97. 



83-85] Zmial Harmonics 107 

The degree of divergenoe of the series (3) when 

y-a-^<0, 
as X approaches the value 1, is given by the theorem * 

J^(a,/3,y,x)=(l-a:)y— ^Jf'(y-a,y-Ay,a;). (6) 

Since the latter series will now be convergent when a; = 1, we see that F (a, ft y, x) becomes 
divergent as (1 -a;)^"*"""; more precisely, for values of x infinitely nearly equal to unity, 
we have 

-P ta, AJ. y, ar) - n (a - 1) . n ()3 - 1) ^* ' ^^^ 

ultimately. 

For the critical case where y - a - /3 = 0, 

we may have recourse to the formula 

^^(a,i8,y,x) = ^J^(a + l,/3 + l,y + l,a-), (8) 

which, with (6), gives in the case supposed 

~J'(a,/3,y,ar)=^(l-a;)-i.J^(y-a,y-fty + l,a:) 

= ^ (1 -ar)-i . J^ (a, 3, a +0 + 1, ar) (9) 

The last factor is now convergent when a; = 1, so that F {a, fi, y, x) is ultimately divergent 
as log ( 1 - a;). More precisely we have, for values of x near this limit, 

i-(«,ft«+A.)=--^iL<^j-;)^iog^ (10) 

85. Of the two series which occur in the general expression Art. 84 (2) 
of a zonal harmonic, the former terminates when n is an even, and the latter 
when n is an odd integer. For other values of n both series are essentially 
convergent for values of fi between ± 1, but since in each case we have 
y — a — j3 = 0, they diverge at the limits /i = ± 1, becoming infinite as 

log (1 - ,*«). 

It follows that the terminating series corresponding to integral values of 
n are the only zonal surface-harmonics which are finite over the unit sphere. 
If we reverse the series we find that both these cases (n even, and n odd) are 
included in the formula f 

p u) = l-3.5...(2n-l) f _ n(n-l) 
^nW- i.2.3...n r 2{2n-l)'* 

I « (n-l)(n-2)(n -3) ) 

^ 2.4.(2n-l)(2n-3) ** ...|, ...^i; 

* Fonyth, Differtniial Equation*, Srd ed., London, 1903, o. vi. 

t For n even this corresponds to ^ = ( - )*** - * -^ ~- ' ~ » jB =0 ; whilst for n odd we have 

A =0. B=( -)**"-^> _? — ^' ^:: . See Heine, t. i. pp. 12, 147. 

2. 4 ... (»-l> '^'^ 



108 Irrotational Motion of a Liquid [chap, v 

where the constant factor has been adjusted so as to make P^{ji)^\ for 
/I = 1 *. The formula may also be written 

The series (1) may otherwise be obtained by development of Art. 82 (6), 
which in the case of the zonal harmonic assumes the form 

^- = ^^"^^a^«r- <^) 

As particular cases of (2) we have 

Po(f^) = l, Pi{fi) = f^^ P, (,i) = i {3/^« - 1), Pa (/i) = i (5^» - 3,x). 

Expansions of P^ in terms of other functions of as independent 
variables, in place of fi, have been obtained by various writers. For 
example, we have 

PAcos0) = l-^'^i'^sin^e + ^^'-^^ 

(4) 

This may be deduced from {2)t, or it may be obtained independently by 
putting /Lt = 1 — 22 in Art. 84 (1), and integrating by a series. 

The function P^ (fi) was first introduced into analysis by LegendreJ as the coefficient 
of h^ in the expansion of 

The connection of this with our present point of view is that if <^ be the velocity-potential 
of a unit source on the axis of a; at a distance c from the origin, we have, on Legendre's 
definition, for values of r less than c, 

1 r r^ 
= - + Pi4 + P,l^ + (5) 

Each term in this expansion must separately satisfy V^ =0, and therefore the coefficient 
Pn must be a solution of Art. 84 (1). Since P», as thus defined, is obviously finite for all 
values of /i, and becomes equal to imity for /a = 1, it must be identical with (1). 

* The functions P^. P,, ... Pj were tabulated by Qlaisher, for values of /i at intervals of *01, 
BrU, Am, Beport, 1879, and are reprinted by Dale, Five-Figure Tables,.., London, 1903. 
A table of the same functions for every degree of the quadrant, calculated under the direction 
of Prof. Perry, was published in the PhiL Mag. for Dec. 1891. Both tables are reproduced in 
Byerly's treatise, also by Jahnke and Emde, Funktionentafeln, Leipzig, 1909. The values of the 
first 20 zonal harmonics, at intervals of 5°, have recently been published by Prof. A. Lodge, 
Phil Trans. A, t. ooiiL (1904). 

t Murphy, Elementary Principles of the Theories of Electricity, <fcc., Cambridge, 1833, p. 7. 
[Thomson and Tait, Art. 782.] 

X "Sur Tattraotion des sph^roides homogdnes," Mim. des Savans jStrangers, t. z. (1786). 



86-86] Zonal Harmonies 109 

For values of r greater than c, the correeponding expansion is 

Ice' 
4ir<^=:-+Pij5+P,^ + (6) 

We can hence deduce expressions, which will be useful to us later. Art. 98, for the 
velocity-potential due to a davble-source of unit strength, situate on the axis of x at a dis- 
tance c from the origin, and having its axis pointing from the origin. This is evidently 
equal to 8<^/dc, where <f> has either of the above forms ; so that the required potential is , 
for r <Cy 

-4^(^-^2p4+3P.J-...), (7) 

and for r>c, 

^{Pr^^2P,^^...) (8) 

The remaining solution of Art. 84 (1), in the case of n integral, can be 
put into the more compact form* 

<?n(A^) = i^«(A^)log]^-Z„, (9) 

where Z„ = ^j^ P„_x + g^^^j ^«-8 + (10) 

This function Q„{n) is sometimes called the zonal harmonic 'of the second 
kind.' 

Thus 
<?o(/*) = ilog}^A Q, (/.) = J (3,i« - 1) log ^ - f /^ 

Q, 0*) = iM log 5^ - 1, e8(M) = i(V-3/*)logi^-$M' + f- 

86. When we abandon the restriction as to symmetry about the axis 
of X, we may suppose /S„, if a finite and single- valued function of a>, to be 
expanded in a series of terms varying as cos sw and sin scj respectively. If 
this expansion is to apply to the whole sphere (i.e. from co = to co = 27r), we 
may further (by Fourier's theorem) suppose the values of « to be integral. 
The difEerential equation satisfied by any such term is 

5i{<'-'"'fl + {"<"+'>-r^}«"-« <■> 

If we put iS„ = (l-^«)*%, 

this takes the form 

a -H'*)^^-^{'>+l)l^^^ + in- 8)(n + 8 + l)v = 0, 

♦ This is equivalent to Art. 84 (4) with, for n even, 4 =0, B=( - )***!— i— V - , ; whilst for 

n odd we have i4 = ( - )* ^"+^* 2.4. ..(n-1) ^ ^q g^ ^^^^^ ^ ^ pp j^^^ j^^ 

o , o , , » n 



110 Irrotational Motion of a Liquid [chap, t 

which is suitable for integration by series. We thus obtain 



1.2 



(n - < - 2) (n - <) (n + * + 1) (n + < + 3) ^_ \ 
+ 1.2.3.4 ^ •••} 

(n - ^ - 3) (w -g-l)(n + g + 2)(n4-^ + 4) ) 

+ 1.2.3.4.5 '^ •"•]' "^^^ 

the factor cos «co or sin «a> being for the moment omitted. In the hyper- 
geometric notation this may be written 

S„ = (1 - ii})^'{AF (\8 - Jn, i + J« + in, i, /i«) 

+ BfiF (J + i« - Jn, 1 + J5 + in, f , ^«)}. . . .(3) 

These expressions converge when yi} < 1, but since in each case we have 

the series become infinite as (1 — /**)*"* at the limits f* = ± 1, unless they 
terminate*. The former series terminates when n — « is an even, and the 
latter when it is an odd integer. By reversing the series we can express 
both these finite solutions by the single formula f 

n (/^)-2n(n-5)!n!^^ ^^ T 2 . (2n - 1) ^ 



+ 



(n - g) (n - g - 1) (n - j? - 2) (n - g - 3) „_ 



n— *— 4 



2 . 4 . (2n - 1) (2n - 3) 
On comparison with Ait. 85 (1) we find that 



- ...l. ..(4) 



P„.(M) = (1 -,*«)*• *^^ (6) 

That this is a solution of (1) may of course be verified independently. 
In terms of sin i^, we have 

P • (cos 0) = ^vT — Vi~i sin' ^ U - , ; , , V sm2 i^ 

" ^ ' 2' (n — «) ! 5 ! ( 1 . (« + 1) * 

(n - j? - 1) (n - g) (n + g + 1) (n + g + 2) . - ^ _ 



+ ' 1.2.(. + l)(. + 2) 8mM^-...|...(6) 

This corresponds to Art. 85 (4), from which it can easily be derived. 

* Rayleigh, Theory of Sound, London, 1877, Art. 338. 

t There are great yarieties of notation in connection with these * associated functions,* as 
they have been called. That chosen in the text was proposed by F. Neumann; and is adopted 
by Whittaker, p. 231. 



86-87] Tesseral and Sectorial Harmonics 111 

Collecting our results we learn that a surface-harmonic which is finite 
over the unit sphere is necessarily of integral order, and is further expressible, 
if n denote the order, in the form 



tf»n 



8^ = A^P^ (ft) + 2 (^,cos 8w 4- 5, sin «cu) P«» (/n), (7) 

containing 2w + 1 arbitrary constants. The terms of this involving co are 
called 'tesseral' harmonics, with the exception of the last two, which are 
given by the formula 

(1 — /i*)*^ {An cos nco + -B„ sin «a>), 

and are called 'sectorial' harmonics*; the names being suggested by the 
forms of the compartments into which the unit sphere is divided by the nodal 
lines Sn = 0. 

The formula for the tesseral harmonic of rank s may be obtained otherwise 
from the general expression (6) of Art. 82 by making n — s out of the n poles 
of the harmonic coincide at the point = of the sphere, and distributing 
the remaining s poles evenly round the equatorial circle 5 = Jtt. 

The remaining solution of (1), in the case of n integral, may be put in 

the form 

S„ = (-4, cos «co + B^ sin «co) Q„« (/x), (8) 

wheret <?.Mf^) == (1 - M*)*'^^^^ (9) 

This is sometimes called a tesseral harmonic 'of the second kind.' 

87. Two surface-harmonics >S, S' are said to be 'conjugate' when 

JJSS'dnj = 0, (1) 

where Sto is an element of surface of the unit sphere, and the integration 
extends over this sphere. 

It may be shewn that any two surface-harmonics, of different orders, 
which are finite over the unit sphere, are conjugate, and also that the 2n -h 1 
harmonics of any given order n, of the zonal, tesseral, and sectorial types 
specified in Arts. 85, 86, are all mutually conjugate. It will appear, later, 
that the conjugate property is of great importance in the physical applications 
of the subject. 

Since Sto = sin &S0Sa> = — S/xSeo, we have, as particular cases of this 
theorem, 

' P^(fi)dfjL = 0, (2) 



/: 



-1 

ri 



j^^P^(fJL),P,{,JL)dfl=0, (3) 



* The prefix 'spherical* is implied; it is often omitted for brevity. 

t A table of'the fanctions Q^ {ft), 0»* (m). for various values of n and s, is given by Brj'an, 
Proc, Camb, PkU. Soc, t. vi p. 297. 



112 Irrotational Motion of a Liquid [chap, v 



and 



j[^P„' (jj,) . P„' (ji) d(i = 0. (4) 



provided m, n are unequal. 

For m = n, it may be shewn* that 



/ 



V,W)'«.-2j^ (6) 



88. We may also quote the theorem that any arbitrary function 
f(fAyO)) of the position of a point on the unit sphere can be expanded in 
a series of surface-harmonics, obtained by giving n all integral values from 
to 00 , in Art. 86 (7). The formulae (5) and (6) are useful in determining 
the coefficients in this expansion. 

Thus, in the case of symmetry about an axis^ the theorem takes the form 

/(/x) = Co + C^P, (h) + C^P^ (/*)+...+ C^Pn (a^) + (7) 

If we multiply both sides by Pn (fi) dfi, and integrate between the limits ±1, 
we find 

co=ir /o*)««/*. (8) 

J —X 

and, generally, 

Cn = ^^j[jWPnWdfl (9) 

For the analytical proof of the theorem recourse must be had to the 
special treatises f; the physical grounds for assuming the possibility of this 
and other similar expansions will appear, 'incidentally, in connection with 
various problems. 

89. Solutions of the equation V^<f> = may also be obtained by the usual 
method of treating linear equations with constant coefficients {. Thus, the 
equation is satisfied by 

or, more generally, by <f> =/(a» + jSy + yz), (1) 

provided a* + jS* + y* = (2) 

* Ferrers, p. 86; Whittaker, pp. 208, 232. 

t For an account of the more recent investigations of the question, see Wangerin, l.c, 

X Forsyth, Differential Equations, p. 444 



87-89] Integral Formulae 113 

For example, we may put 

a, ^, y = 1, i cos ^, isin ^, (3) 

or, again, a, j8, y = 1, i cosh w, sinh u (4) 

It may be shewn* that the most general solution possible can be obtained 
by superposition of solutions of the type (1). 

Using (3), and introducing the cylindrical co-ordinates x, ro, a>, where 

y = m cos o), 2 = ro sin a>, (5) 

we build up a solution symmetrical about the axis of x if we take 

1 r*» 

<^ = — j f{x + im cos (& — o))} ^. 

For, since the integration extends over a whole circumference, it is immaterial 
where the origin of ^ is placed, and the formula may therefore be written f 

<f> = crl /(aJ4-iwcos&)d^ = - I /(a: + irocosa)(i^. ..(6) ^ 

This is remarkable as giving a value of ^, symmetrical about the axis 
of x^ in terms of its values /(x) at points of this axis!]:. It may be shewn, 
by means of the theorem of Art. 38, that the form of ^ is in such a case 
completely determined by the values over any finite length of the axis§. 

As particular cases of (6) we have the functions 

1 /•» 1 r» 

- I (aj + iro cos ^Y cF^, - I (a? 4- its cos &)-'*-^ da, 

where n will be supposed to be integral. Since these are soUd harmonics 
finite over the unit sphere, and since, for ro = 0, they reduce to r** and r """^, 
they must be equivalent to P„ (fi) r»*, and P„ (fi) r-"-^, respectively. We 
thus obtain the forms 

^n (ft) = - r^ + *V(1 - M*) cos a}« (i&, (7) 



p 



. X __ 1 r" ^ _ _ _ 

^^^ ~ 77 / n lu + CJ(l^ U*i COS ^>+i' ^^ 



*• "*' 77 ./ (m H- i \/(l - A^*) COS ^}«+i ' 

due originally to Laplace || and Jacobi^, respectively. 

• Whittaker, Month, Not. R, AM. Soc. t. Ixii. (1902). 

t Whittaker, Modern Analysis, c. xiii. 

t Whittaker, p. 321. 

§ Thomson and Tait Art. 498. 

I! MSc. Ca. Livre 11—, c. U. 

t CreUe, t. xxtL (1843) [OesammeUe Werke, Berlin, 1881. . ., t. vi. p. 148], 

L. H. g 



114 Irrotdtional Motion of a Liquid [ceiap. v 

90. As a first application of the foregoing theory let us suppose that an 
arbitrary distribution of impulsive pressure is applied to the surface of a 
spherical mass of fluid initially at rest. This is equivalent to prescribing an 
arbitrary value of <f> over the surface ; the value of <f> in the interior is thence 
determinate, by Art. 40. To find it, we may suppose the given surface- value 
to be expanded, in accordance with the theorem quoted in Art. 88, in a series 
of surface-harmonics of integral order, thus 

^ = /So + 5i + /Sj+ ... +/S„+ (1) 

The required value is then 

<^ = So + ^iS, + ^,S,+ ...-f JiS„+..., (2) 

for this satisfies V^ = 0, and assumes the prescribed form (1) when r = a^ the 
radius of the sphere. 

The corresponding solution for the case of a prescribed value of <^ over 
the surface of a spherical cavity in an infinite mass of Uquid initially at rest 
is evidently 

<A = ^% + ^«i + ^'««+ •.+^>n+ (3) 

Combining these two results we get the case of an infinite mass of fluid 
whose continuity is interrupted by an infinitely thin vacuous stratum, of 
spherical form, within which an arbitrary impulsive pressure is applied. The 
values (2) and (3) of ^ are of course continuous at the stratum, but the 
values of the normal velocity are discontinuous, viz. we have, for the internal 
fluid, 

dr a ' 

and for the external fluid 

^ = -2(«+l)^. 
or ^ ' a 

The motion, whether internal or external, is therefore that due to a 
distribution of simple sources with surface-density 



over the sphere ; see Art. 58. 



2(2«+l)|' (4) 



91. Let us next suppose that, instead of the impulsive pressure, it is the 
normal velocity which is prescribed over the spherical surface ; thus 

|^ = S,-hS,+ ...+S„+ ..., (1) 



90-92] Applications of Spherical Harmonics 115 

the term of zero order being necessarily absent^ since we must have 

jjfr'^'^' (2) 

on account of the constancy of volume of the included mass. 
The value of </> for the internal space is of the form 

<f> - AjtS^ + ^2^*^2 + . . . + A^r'^S^ + . . . , (3) 

for this is finite and continuous, and satisfies V^^ = 0, and the constants can 
be determined so as to make d<f>ldr assume the given surface- value (1) ; viz. 
we have nAnd^-^ = 1. The required solution is therefore 

^ = a2-^iS„ (4) 

The corresponding solution for the external space is found in like manner 
to be 

'^^-'^^dnr-S^- (5) 

The two solutions, taken together, give the motion produced in an 
infinite mass of liquid which is divided into two portions by a thin spherical 
membrane, when a prescribed normal velocity is given to every point of the 
membrane, subject to the condition (2). 

The value of <f> changes from dZSn/n to — al,SJ{n + 1), as we cross the 
membrane, so that the tangential velocity is now discontinuous. The motion, 
whether inside or outside, is that due to a double-sheet of density 

_a2^^V^^-; (6) 

n(n4- 1) 

see Art. 58. 

The kinetic energy of the internal fluid is given by the formula (4) of 
Art. 44, viz. 

2T = pjf<f>^dS = pa'lljfs^Hw, (7) 

the parts of the integral which involve products of surface-harmonics of 
different orders disappearing in virtue of the conjugate property of Art. 87. 

For the external fluid we have 

2T^-pjj<f>f^dS = pa»l^Jfs„^dm (8) 

92. A particular, but very important, case of the problem of the 
preceding Article is that of t he motio n nf t\ nnti"^ wrhfiffi in **? infinite 
mass of liquid which is at rest at infinity^ If we take the origin at the 
centre of the sphere, and the axis of x in the direction of motion, the 

8—2 



116 Irrotational Motion of a Liquid [chap, v 

normal velocity at the surface is TJxjr, = U cos d, where U is the velocity 
of the centre. Hence the conditions to determine <f> are (1°) that we must 
have V^^ = everywhere, (2°) that the space-derivatives of <f> must vanish at 
infinity, and (3°) that at the surface of the sphere (r = a) we must have 

-^=Ucosd (1) 

I 

The form of this suggests at once the zonal harmonic of the first order ; we 
therefore assume 

The condition (1) gives — 2i4/a* = Z7, so that the required solution is* 

<f>^\u'^oo^e (2) 

It appears on comparison with Art. 56 (4) that the motion of the fluid is 
the same as would be produced by a dauble-source of strength %TVa^, situate 
at the centre of the sphere. For the forms of the stream-lines see p. 122. 

To find the energy of the fluid motion we have 

2T = - /> /J<^ ^ dS = \paV^ r COS* .27Ta 8m 6 .add 

= inpa^U^ = M'U^ (3) 

P^y'3 ii M* = §7r/3a'. It appears, exactly as in Art. 68, that the effect of thie fluid 
pressure is equivalent simply to an addition to the inertia of the solid, the 
increment being now half the mass of the fluid displaced f. 

Thus in the case of rectilinear motion of the sphere, if no external forces 
act on the fluid, the resultant pressure is equivalent to a force 

-«'f w 

in the direction of motion, vanishing when U is constant. Hence if the 
sphere be set in motion and left to itself, it will continue to move in a 
straight line with constant velocity. 

The behaviour of a solid projected in an actual fluid is of course quite 
different; a continual application of force is necessary to maintain the 
motion, and if this be not supplied the soUd is gradually brought to rest. 

* Stokes, "On some oases of Fluid Motion," Camb. Trans, t. Tiii. (1843) [Papers, t. i. p. 17]. 

Diriohlet, *'Ueber die Bewegung eines festen Korpers in einem inoompressib^ fliissigen 
Medium,'' Berl Monatsber. 1852 [Werle, Berlin, 1889-97, t. ii. p. 115]. 

t Stokes, l.e. The result had been obtained otherwise, on the hypothesis of infinitely 
small motion, by Green, "On the Vibration of Pendulums in Fluid Media," Edin. Trans. 1833^ 
[Papers, p. 315]. 



92-93] Motimi of a Sphere 117 

It must be remembered however, in making this comparison, that in a 
* perfect* fluid there is no dissipation of energy, and that if, further, the fluid 
be incompressible, the solid cannot lose its kinetic energy by transfer to the 
fluid, since, as we have seen in Chapter iii., the motion of the fluid is entirely 
determined by that of the soUd, and therefore ceases with it. 



5=^-i«*+^(0 (8) 



If we wish> to verify the preceding results by direct calculation from the formula 

P 

we must remember, as in Art. 68, that the origin is in motion, and that the values of r 
and for a fixed point of space are therefore increasing at the rates - U cos $, and 
((/sin ^)/r, respectively. We thus find, for r^a, 

?=Ja^coe^+T^l7«co8 2^-^?7«+JF'(0 (6) 

The last three terms are the same for surface-elements in the positions 6 and ir -^; so 
that, when U is constant, the pressures on the various elements of the anterior half of the 
sphere are balanced by equal pressures on the corresponding elements of the posterior half. 
But when the motion of the sphere is being accelerated there is an excess of pressure on 
the anterior, and a defect on the posterior half. The reverse holds when the motion is 
being retarded. The resultant effect in the direction of motion is 



-/; 



JTJ 

2ira sin $ .ad$ , p cos $= - §"'pa' -^, 



as before. 



93. The same method can be applied to find the motion produced in a 
Uquid contained between a solid sphere and a fixed concentric spherical 
boundary, when the sphere is moving with given velocity U. 

The centre of the sphere being taken as origin, it is evident, since the 
space occupied by the fluid is Umited both externally and internally, that 
soUd harmonics of both positive and negative degrees are admissible ; they 
are in fact required, in order to satisfy the boundary conditions, which are 

— d<l>fdr = U cos 6, 

for r = a, the radius of the sphere, and 

d<f>/dr = 0, 

for r = b, the radius of the external boundary, the axis of x being as before in 
the direction of motion. 

(B\ 
Ar + -g j cos 6y (1) 

and the conditions in question give 

a' 6' 
whence ^^^s^' ^^^^^.U (2) 



118 Irrotational Motion of a Liquid [chap, v 

The kinetic energy of the fluid motion is given by 

the integration extending over the inner spherical surface, since at the outer 
we have 3<^/3r = 0. We thus find 

2r = ^^±^V'C^'. (3) 

It appears that the effective addition to the inertia of the sphere is now* 

6» + 2a« 



TT 



6» - a» 



/>a« (4) 



As h diminishes from oo to a, this increases continually from ^irpa^ to oo , in 
accordance with Lord Kelvin's minimum theorem (Art. 45). In other words, 
the introduction of a rigid spherical partition in the problem of Art. 92 acts 
as a constraint increasing the kinetic energy for any given velocity of the 
sphere, and so virtually increasing the inertia of the system. 

94. In all cases where t he motion of a liquid takes pl ace in a series o f 
pla nes passing through a common line, and is Jhej same i n e ach such plane , 
there exists a f^reft^-jt^nnfyhif^ panaToptviiflln some of its properties to the two- 
dimensional stream-function of the last Chapter. If in any plane through 
the axis of symmetry we take two points A and P, of which A is arbitrary, 
but fixed, wtile P \a variable, then considering the annular surface generated 
by any line AP, it is plain that the flux across this surface is a function of 
the position of P. Denoting this function by 27t^, and taking the axis of x 
to coincide with that of symmetry, we may say that is a function of x and 
tD, where x is the abscissa of P, and m, = (y* + «*)% is its distance from the 
axis. The curves ^ = const, are evidently stream-lines. 

If P' be a point infinitely near to P in a meridian plane, it follows from 
the above definition that the velocity normal to PP' is equal to 

27r8^ 
27rtD.PP" 

whence, taking PP' parallel first to m and then to x, 

where u and v are the components of fluid velocity in the directions of x and 
w respectively, the convention as to sign being similar to that of Art. 59. 

These kinematical relations may also be inferred from the form which the 
equation of continuity takes under the present circumstances. If we express 

♦ Stokes, Lc. ante p. 116. 



93-94] Stokes' Stream-Function 119 

that the total flux into the annular space generated by the reyolution of an 
elementary rectangle SxSm is zero, we find 

^ {u . ^mhw) See + X- (u . ^TTwhx) hm = 0, 

or 5- (mu) -I- ^r- Iwv) = 0, (2) 

which shews that wv .dx — mu . dm 

is an exact differential. Denoting this by d*ff we obtain the relations (1)*. 

So far the motion has not been assumed to be irrotational ; the condition 
that it should be so is 

dv 9^ _ /^ 

dx 9t5 ~ ' 

whichleadsto P^ + ft ]l^ = o (3) 

ox^ ow^ woto 

The differential equation of <f} is obtained by writing 

„=_?^ „=_?^ 

dx' dm 

"P).-it» S + ^ + sE-o W 

It appears that the functions ^ and are not now (as they were in Art. 62) 
interchangeable. They are, indeed, of different dimensions. 

The kinetic energy of the liquid contained in any region bounded by 
surfaces of revolution about the axis is given by 



-X" . 27Tmd8 

ZDOS 



-=27TpJ4>tb/, (5) 

8^ denoting an element of the meridian section of the bounding surfaces, and 
the integration extending round the various parts of this section, in the 
proper directions. Compare Art. 61 (2). 

* The stream-function for the case of symmetry about an axis was introduced in this manner 
by Stokes. "On the Steady Motion of Incompressible Fluids," Camb. Trans, t. vii. (1842) [Papers, 
t. i p. 1]. Its analytical theory has been treated very fully by Sampson, "On Stokes' Current- 
Function," PhU. Trans, A, t. clxxxii. (1891). 



120 Irrotational Motion of a Liquid [chap, v 

95. The velocity-potential due to a point-source at the origin is of the 
form 

<!> = ] (1) 

The flux thrdugh any closed curve is in this case numerically equal to the 

solid angle which the curve subtends at the origin. Hence for a circle with 

Ox as axis, whose radius subtends an angle d at 0, we have, attending to the 

sign, 

2w}t = - 27r (1 - cos fl). 

Omitting the constant term we have 

^ = r = ai (2) 

The solutions corresponding to any number of simple sources situate at 
various points of the axis of x may evidently be superposed; thus for the 
double-source 

^ = -air = T« ' (^) 

, , av nj2 sin^fl 
^« l^ave = _ g^^ = _ _ = - -_ (4) 

And, generally, to the zonal solid harmonic of degree — w — 1, viz. to 

a** 1 

corresponds* == ^ g^^-^ (6) 

A more general formula, applicable to harmonics of any degree, fractional 
or not, may be obtained as follows. Using spherical polar co-ordinates r, d, 
the component velocities along r, and perpendicular to r in the plane of the 
meridian, are found by making the linear element PP' of Art. 94 coincide 
successively with rh6 and 8r, respectively, viz. they are 

f sin fl r3fl' r sin fl 3r ^ ' 

Hence in the case of irrotational motion we have 

~^m-"t |--»l <«) 

Thus if ^ = f«Sn, (9) 

where Sn is a zonal harmonic of order n, we have, putting /x = cos 0, 

* Stefan, "Ueber die Kraftlinien eines urn eine Axe symmetrischen Feldes," Wied, Ann, 
t. zvii (1882). 



95-96] Streani'Lhies of a Sphere 121 

The latter equation gives 

^ = «il'""(l-'*')f' : <1<^) 

which must necessarily also satisfy the former; this is readily verified by 
means of Art. 84 (1). 

Thus in the case of the zonal harmonic P„, we have as corresponding 
values 

^ = r-P,(/*), ^=-^r»+Ml-M')^". (11) 

and ^ = r— ip„(/x), ^ = - 1 r- (1 - /*«) '^f" (12) 

of which the latter must be equivalent to (5) and (6). The same relations 
hold of course with regard to the zonal harmonic of the second kind, Q„ . 

96. We saw in Art. 92 that the motion produced by a solid sphere in 
an infinite mass of liquid may be regarded as due to a double-source at 
the centre. Comparing the formulae there given with Art. 95 (4), it appears 
that the stream-function due to the sphere is 

= - i Z7 - sin2 e ( 1 ) 

The forms of the stream-lines corresponding to a number of equidistant 
values of t/f are shewn on the next page. The stream-Unes relative to the 
sphere are figured in a diagram near the end of Chapter vil^*23« # 

Again, the stream-function due to two double-sources having their axes 
oppositely directed along the axis of x will be of the form 

^ = VY -7-8-' (2) 

where r^, r^ denote the distances of any point from the positions, P^ and Pj, 
say, of the two sources. At the stream-surface = we have 

i.e. the surface is a sphere in relation to which P^ and P^ are inverse points. 
If be the centre of this sphere, and a its radius, we find 

AjB = OPiVa* = a^lOP^^ (3) 

This sphere may be taken as a fixed boundary to the fluid on either side, and 
we thus obtain the motion due to a double-source (or say to an infinitely 
small sphere moving along Ox) in presence of a fixed spherical boundary. 
The disturbance of the stream-Unes by the fixed sphere is that due to a 
double-source of the opposite sign placed at the 'inverse' point, the ratio of 



122 



Irrotational Motion of a Liquid 



[chap. V 



the strengths being given by (3)*. This fictitious double-source may be 
called the 'image' of the original one. 




97. Rankinef employed a method similar to that of Art. 71 to discover 
forms of solids of revolution which will by motion parallel to their axes 
generate in a surrounding liquid any given type of irrotational motion 
symmetrical about an axis. 

The velocity of the solid being U, and 85 denoting an element of the 
meridian, the normal velocity at any point of the surface is Udm/ds, and that 

* This result was given by Stokes, "On the Resistance of a Fluid to two Oaoillating Spheres," 
Brit Ass. Report, 1847 [Papers, t. i. p. 230]. 

t **0n the Mathematical Theory of Stream Lines, especially those with Four Foci and 
upwards," PhiL Trans. 1871, p. 267 (not included in the collection referred to on p. 61 ante). 



96^98] Motion of Two Spheres 123 

of the fluid in contact is given by — dt/t/wds. Equating these and integrating 
along the meridian, we have 

= — iUw^ + const. ^ (1) 

If in this we substitute any value of t// satisf jnng Art. 94 (3), we obtain the 
equation of the meridian curves of a series of solids, each of which would by 
its motion parallel to x give rise to the given system of stream-lines. 

In this way we may readily verify the solution already obtained for the 
sphere; thus, assuming 

^^Aw^/r^, (2) 

we find that (1) is satisfied for r = a, provided 

A=^^^Ua\ : (3) 

which agrees with Art. 96 (1). 

98. The motion of a liquid bounded by two spherical surfaces can be 
found by successive approximations in certain cases. For two soUd spheres 
moving in the line of centres the solution is greatly faciUtated by the result 
given at the end of Art, 96, as to the 'image' of a double-source in a fixed 
sphere. 

Let a, 6 be the radii, and c the distance between the centres A, B. Let U be the 
velocity of A towards B, V that of B towards A, Also, P being any point, let AP=r, 
BP =/, PAB = 0, PBA =e\ The velocity-potential will be of the form 

u<t> + u'4>% (1) 

where the functions and <f>' are to be determined by the conditions that 

V^<t> =0, ^V =0, (2) 

P 



* * 




^ ir. I V"ff, 



throughout the fluid, that their space-derivatives vanish at infinity, and that 



over the surface of A, whilst 



^=-co8tf. ^'=0. (3) 



^,=0. g:=— ' (4) 



over the surface of B. It is evident that is the value of the velocity-potential when A 
moves with unit velocity towards B, while B \b At rest; and similarly for 0'. 



124 Irrotational Motion of a Liquid [ohap. v 

To find 0, we remark that if B were absent the motion of the fluid would be that due 
to a certain double-source at A having its axis in the direction AB, The theorem of Art. 96 
shews that wc may satisfy the condition of zero normal velocity over the surface of B 
by introducing a double-source, viz. the 'image* of that at ^ in the sphere B» This image 
is at Hi, the inverse point of A with respect to the sphere B\ its axis coincides with AB^ 
and its strength is - yijt^lc^, where /zq is the strength of the original source at A, viz. 



/^ 



= 2ira'. 



The resultant motion due to the two sources at A and H^ will however violate the condi- 
tion to be satisfied at the surface of the sphere Ay and in order to neutralize the normal 
velocity at this surface, due to H^, we must superpose a double-source at H^, the image 
of H^ in the sphere A, This will introduce a normal velocity at the surface of B, which 
may again be neutralized by adding the image of H2 in B, and so on. If /ix» f4» Ms* • • • be 
the strengths of the successive images, and/^, f2*h» • • • t^eir distances from Ay we have 



/x=c-- . 


^' "A • 




Ml" fi" 




f-"* 
^*-/,' 




M, /," 








M6~ /."'/ 



cuid so on, the laws of formation being obvious. The images continually diminish in 
intensity, and this very rapidly if the radius of either sphere is small compared with the 
shortest distance between the two surfaces. 

The formula for the kinetic energy is 

2T=-p I l{U<t> + U'<t>') (^^+U' ^^ d8=LU^ +2MUU' +NU'^ (6) 

provided 

where the suffixes indicate over which sphere the integration is to be effected. The 
equality of the two forms of M follows from Green's Theorem (Art. 44). 

The value of near the surface of A can be written down at once from the results (7) 
and (8) of Art. 85, viz. we have 



4^0=(/io+/*a+/i4 + ...)^^-2fj| + ^+ •..Vcos^+&o., 



(8) 



the remaining terms, involving zonal harmonics of higher orders, being omitted, as they 
will disappear in the subsequent surface-integration, in virtue of the conjugate property of 
Art. 87. Hence, putting 8</>/8n = - cos B, we find with the help of (5) 

^ = ip(;*o+3/ia+3Am-...) = }7r^»(^l-f3^g+3 ^^^3^^_^^^^3^^3 -h...j. ...(9) 

It appears that the inertia of the sphere Jl is in all cases increased by the presence of 
a fixed sphere B. Compare Art. 93. 

The value of N may be written down from symmetry, viz. it is 

.V =S.p6» (1 +3^1'^ +3 ^^^,, ^f_^J^,^^ -^ + . . . ) (10) 



98-99] 



Motion of Two Spheres 



125 



where 



//— ? . 



f'-c "' 



ye — f f* ^ 



(11) 



and so on. 



To calculate if we require the value of if/ near the surface of the sphere A ; this is due 
to double-sources ij^\ /aj', /x,', /ig', ... at distances c, c -/i', c -/g', c -/g', . . . from A^ where 
/iq' = - 2ir6^, and 



Ms 







«» 




(c -/,')" 







(12) 



Hence 



and so on. This gives, for points near the surface of A, 
4ir*' =(/h' +/^' +M5' + . . "^7/ - 2 (^ + ^^^^y^ 

_« o86»/ a36» a«6« 1 

When the ratios a/c and hjc are both small we have 

L = t7rpa3(l+3^), M^2np^, i^^j^pfts ^1 +3^^^ (15) 

approximately ♦. 

If i^ the preceding results we put h=a, U' = U, the plane bisecting AB at right angles 
will be a plane of symmetry, and may therefore be taken as a fixed boundary to the fluid 
on either side. Hence, putting c =2h, we find, for the kinetic energy of the liquid when a 
sphere is in motion perpendicular to a rigid plane boundary, at a distance h from it. 



2r=|«-pa»(l+ip + ...)c7* (16) 



a result due to Stokes. 



/ 



99. When the spheres are moving at right angles to the line of centres 
the problem is more diffictdt; we shall therefore content ourselves with the 
first steps in the approximation, referring, for a more complete treatment, to 
the papers cited below. 



* To this degree of approximation the results may be more easily obtained without the use 
of * images,* the procedure being similar to that of the next Art. 



126 In^otational Motion of a Liquid [chap, v 

Let the spheres be moving with velocities F, F' in i)arallel directions at right angles to 
Ay Bj and let r, B, <a and r', 6\ <»' be two systems of spherical polar co-ordinates having their 
origins at A and B respectively, and their polar axes in the directions of the velocities 
Vi F^ The velocity-potential will be of the form 

F« + V'ft>\ 
with the surface-conditions 

5^=-cos^, "i"=^» forr=o, (1) 

and 5^=0, ^=-cos^', forr'=6 •. (2) 

or or ^ ' 

If the sphere B were absent the velocity-potential due to unit velocity of A would be 

J -^ cos ^. 

Since r cos 6=r^ cos 0", the value of this in the neighbourhood of B will be 

J ^ r' cos ^, 

approximately. The normal velocity at the surface of B, due to this, will be canceUed by 
the addition of the term 

a'ft* cos B' 

which, in the neighbourhood of A becomes equal to 

I —5- r cos 6, 

nearly. To rectify the normal velocity at the surface of ^, we add the term 
Stopping at this points and collecting our results, we have, over the surface of A, 

0-i«(l+i^)coB^, (3) 

a' 
and at the surface of B, ^ =1^ * is ^^ ^ (^) 

Hence if we denote by P, Q, R the coefficients in the ezpreraion for the kinetic energy, 
viz. 

2T=PV* +2QVV' +Rr'* (5) 

we have F= - p JJ^grfS^ =f^pa»(l + J^/ 






(6) 



99-100] Cylindrical Harmonics 127 

The oase of a sphere moving parallel to a fixed plane boundary, at a distance A, is 
obtained by putting 6 =a, V = V\c =2A, and halving the consequent value of T; thus 

2r=|,r/K.»(l + A^!)F* (7) 

This result, which was also given by Stokes, may be compared with that of Art. 98 (16)*. 



Cylindrical Harmonics. 

100. In terms of the cylindrical co-ordinates x, to, co introduced in 
Art. 89, the equation V*^ = takes the form 

a*^ av.i?^ ,l!V = o m 

dx^ 9nj* w dm ro* 9co* 

This may be obtained by direct transformation, or more simply by expressing 
that the total flux across the boundary of an element hx ,hm . xnh<i} is zero, 
after the manner of Art. 83. 

In the case of symmetry about the axis of x^ the equation reduces to the 
form (4) of Art. 94. A particular solution is then if> = 6*** x (^)» provided 

X" (t") + ^ X' («>) + **X («) = (2) 

This is the differential equation of 'Bessel's Functions' of zero order. Its 
complete primitive consists, of course, of the sum of two definite functions 
of tzj, each multipUed by an arbitrary constant. That solution which is finite 
f or nj = is easily found in the form of an ascending series ; it is usually 
denoted by CJ© (^)> where 

.7o(0 = l-|| + 2i^4,- (3) 

We have thus obtained solutions of V«^ = of the types t 

^ = g*t« J^ (km) (4) 

It is easily seen from Art. 94 (1) that the corresponding value of the 
stream-function is 

= If lue*** Jo' i]cm) (5) 

* For a fuUer analytical treatment of the problem of the motion of two spheres we refer to 
the following papers: W. M. Hicks, '*0n the Motion of two Spheres in a Fluid," Phil. Trawt, 
1880, p. 455; R. A. Herman, "On the Motion of two Spheres in Fluid,** Quart. Jowm, Math, 
t. zxii. (1887); Basset, ''On the Motion of Two Spheres in a Liquid, fto." Proc. Lond. Math. 
8oc. t. zviii. p. 360 (1887). See also C. Neumann, Hydrodynamische Untersuehungen, I^ipzig, 
1883; Basset, Hydrodynamics, Cambridge, 1888, t. i. The mutual influence of 'pulsating' 
spheres, ue, of spheres which periodically change their volume, has been studied by C. A. Bjerknes, 
with a view to a mechanical illustration of electric and other forces. A full account of these 
researches is givrai by his son Prof. V. Bjerknes in his VorUsungen Hber hydrodynamische Femkrafte, 
Leipzig, 1000-1902. The question is also treated by Hicks, Camb. Proc. t. iii. p. 276 (1870). 
t. iv. p. 29 (1880), and by Voigt, (?dtt. Nachr. 1891, p. 37. 

t Except as to notation these solutioni* are to be found tn Poisson, l.c. ante p. 17. 



128 Irrotatwiial Motion of a Liquid [chap, v 

The formula (4) may be recognized as a particular case of Art. 89 
(6); viz. it is equivalent to 






g±*(*+.-irco8^) ^^ (6) 



smce 



J^(^) = l rcos(Scos^)ci&=- [^ e'^^^g^, (7) 

as may be verified by developing the cosine, and integrating term by term. 

Again, (4) may also be identified as the Hmiting form assumed by a 
spherical solid zonal harmonic when the order (n) is made infinite, provided 
that at the same time the distance of the origin from the point considered be 
made infinitely great, the two infinities being subject to a certain relation*. 

Thus we may take 

^ = ^„P„(/.) = (l + ?)"ff,(«,) (8) 

where we have temporarily changed the meanings of x and to, viz. 

r = a-\-x^ to = 2a sin |fl, 
whilst 

U /n,\ - 1 _ ^ (^ + ^) ^* . (n ~ 1) n (n + 1) (n + 2) ^ . 

see Art. 85 (4). If we now put h = n\a^ and suppose a and n to become 
infinite, whilst h remains finite, the symbols x and to will regain their former 
meanings, and we reproduce the formula (4) with the upper sign in the 
exponential. The lower sign is obtained if we start with 

The same procedure leads to an expression of an arbitrary function of m 
in terms of the Bessel's Function of zero order f. According to Art. 88, an 
arbitrary function of latitude on the surface of a sphere can be expanded in 
spherical zonal harmonics, thus 

J- (,*) = 2 (« + i) P„ (/Lt) J^^ J (/) P„ 1^) dfl' (10) 

If we denote by w the length of the chord drawn to the variable point 
from the pole (0 = 0) of the sphere, we have 

t!j^2asin^d, toSto = — a^8/x, 
where a is the radius, so that the formula may be written 

fivj) = ], 2 (n + i) H„ (ID) rf(w') H, (w') w'dm' (11) 

* This process was indicated, without the restriction to symmetry, by Thomson and Tait, 
Art. 783 (1867). 

f The procedure appears to be due substantially to C. Neumann (1862) ; for the history of the 
theorem (12) see Heine, t. i p. 442, and Niekien (op. eU, p. 129), p. 360. 



100-101] Cylindrical Harmonics 129 

If we now put * = -, SA = -, 

and finally make a infinite, we obtain the important theorem : 

/ (to) = r Jo (km) hdkT f (tn') J© (km') w'dw' (12) 

JO Jo 

101. If in (1) we suppose ^ to be expanded in a series of terms varying 
as cos^cu or sin^co, each such term will be subject to an equation of 
the form 

a^ + aS-^ + SaS-Si^""^ ^^^^ 

This will be satisfied by ^ = e*** x (^)» provided 

x"M+^x'M + (a«-^)xM = 0, (14) 

which is the differential equation of Bessel's Functions of order 8*. The 
solution which is finite for to = may be written x (*) = C'J, (tro), where 

J^ (0 = 2^ n"(7) r " 2 (2« + 2) "^ 2 . 4 (2« -I- 2) (2« + 4) ""•••}•• -(1^) 

The complete solution of (14) involves, in addition, a Bessel's Function 
'of the second kind' with whose form we shall be concerned at a later period 
in our subjectf. 

We have thus Obtained solutions of the equation V*^ = 0, of the types 

^ = 6*»« J. (Jfcm) ^.^1 sa> (16) 

These may also be obtained as limiting forms of the spherical solid harmonics 

^'* » w \ cos) a«+i 7> . / X cos] 

with the help of the expansion (6) of Art. 86 J . 

* Forsyth, Art. 100; Whittaker, o. zii 

t For the further theory of the Bessel's Functions of both kinds recourse may be had to 
Lommel, Studien ueber die BeawPtehen FunkHonenp Leipzig, 1868; Gray and Mathews, Treatise 
on Bessel Functions, London, 1896 ; H. Weber, PartieUe Differentidlgleiehungen d. nuUh. Physik, 
Braunschweig, 1900-01; Nielsen, Handbueh d, Theorie d. Cylinderfunkiionen, Leipzig, 1904; 
and to the treatises of Heine, Todhunter, Forsyth, Byerly, and Whittaker, already cited. An 
ample account of the subject, from the physical point of view, will be found in Rayleigh's Theory 
of Sound, oc. ix., zviii., with many important applications. 

Numerical tables of the functions J, (^) have been constructed by Bessel and Hansen, and more 
recently by Meissel {BerL Ahh, 1888). Hansen's tables are reproduced by Lommel, and (partially) 
by Rayleigh and Byerly; whilst Meissel's tables have been reprinted by Gray and Mathews. 
Abridgments to five and four figures respectively are included in the collections of Dale and of 
Jahnke and Emde referred to on p. 108. 

X The connection between spherical surface-harmonics and Bessel's Functions was noticed by 
Mehler," Ueber dieVertheilung d. statischen Elektricitat in einem v. zwei Kugelkalotten begrenzten 

L.H. 9 



130 Irrotational Motion of a Liquid [chap, v 

102. The formula (12) of Art. 100 enables us to write down expressions, 
which are sometimes convenient, for the value of <f> on one side of an infinite 
plane {x = 0) in terms of the values of <l> or 3^/3n at points of this plane, in 
the case of symmetry about an axis (Ox) normal to the plane*. Thus if 

^ = J? (id), for a; = 0, (1) 

we have, on the side x > 0, 

^ = r e-*« Jo (*C7) UhT F (tn') Jo (*^) ^'dm' (2) 

Again, if • "" af "" -^^^^^ for « = 0, (3) 

we have <f> = I 6-** Jq {*ro) dk I / (to') Jo {Jew') w'dw' (4) 

The exponentials have been chosen so as to vanish for cr = oo . 

Another solution of these problems has already been given in Art. 58, 
from equations (12) and (11) of which we derive 

^ = ^-/K©'^' (^) 

respectively, where r denotes distance from the element hS of the plane to 
the point at which the value of ^ is required. 

We proceed to a few applications of the general formulae (2) and (4). 

1°. If, in (4), we assume /(ro) to vanish for all but infinitesimal values 
of ro, and to become infinite for these in such a way that 



and 



/, 



f(m) 27rujdw = i, 




we obtain 4^ = I 6"** J© (kw) dk, (7) 

and therefore, since Jq = — Ji, 

/•oo 

47r0 = - tn I e-*» Ji {kw) dk, (8) 

Jo 

by Art. 100 (5). 

Korper," CreUe, t. Ixviii. (1S68). It was investigated independently by Bayleigb, ''On the 

Relation between the Functions of Laplace and Bessel," Proc. Lond, Math. Soc, t. ix. p. 61 

<1878) [Papers, t. i. p. 338]; see also Theory of Sound, Arts. 336, 338. 

There are also methods of deducing BesseFs Functions 'of the second kind' as limiting 

cos I 
forms of the spherical harmonics Q^ (ft), Q^* (/t) . > «»; for these see Heine, t i. pp. 184, 232. 

* The method may be extended so as to be free from this restriction. 



102] Applications of Cylindrical Harmonics 131 

By comparison with the primitive expressions for a point-source at the 
origin (Art. 95), we infer that 

f e-^ Jo (M d* = J, r «"*« Ji (M dk = ~^-^. , ... .(9) 
Jo T Ja r [r -\- X) 

where r = ^/{x^ 4- to*) ; these are in fact known results*. 

2^. Let us next suppose that sources are distributed with uniform 
density over the plane area contained by the circle tn = a, jr = 0. Using the 
series for Jo, «^i, or otherwise, we find 

I Jo {km) mdm = j^Ji (ka) (10) 

.'0 ^ 

Hence t 

(11) 

where the constant factor has been chosen so as to make the total flux 
through the circle equal to unity. 

3°. Again, if the density of the sources, within the same circle, vary 
as l/\/(^* ~ ®*), w® hskve to deal with the integral I 

/"^ (M:^;?(J^) = »/*Vo (te sin^) sin ^ d^ = "i^. . .(12) 

where the evalustion is effected by substituting the series form of Jq, and 
treating each term separately. Hence 

(13) 

if the constant factor be determined by the same condition as before §. 

It is a known theorem of Electrostatics that the assumed law of density 
makes (f> constant over the circular area. It may be shewn independently 
that 

Jo (kw) sin ia -T- = Jtt, or sin"^ - 



/ 
/ 





00 



Ji (km) Bmka-j- = ^^ -, or — 

/cm m 



(14) 



* The former is due to Lipsohitz, Crelle, t. Ivi p. 189 (1869); Bee Gray and Mathews, p. 72. 
The latter follows by di£ferentiation with respect to w and integration with respect to z, 

t Cf. H. Weber, CreUe, t. Ixxv. p. 88; Heine, t. u. p. 180. 

X The formula (12) has been given by various writers; see Rayleigh, Papers, t. iiL p. 98; 
Hobson, Proc. Land. Math. Soe. t. xxv. p. 71 (1893). 

§ CJf. H. Weber, Crdk, t. Ixxv. (1873); Heine, t. ii p. 192. 

9—2 



132 Irrotational Motion of a Liquid [chap, v 

according as m ^ a*. The formulae (13) therefore express the flow of a liquid 
through a circular aperture in a thin plane rigid wall. Another solution will 
be obtained in Art. 108. The corresponding problem in two dimensions was 
solved in Art. 66, 1°. 

4°. Let us next suppose that when x = 0, we have <f> = C ^/(a* — to*) 
for tD < a, and ^ = for ro > a. We find 

I Jq (km) \/(a* — tD*) tDdm = a^ I Jq {ka sin &) sin S^ cos* S^d& = a'^i (fei), 
Jo Jo 

(15) 

provided 0,(O = i(l-^ + 27A77""-:)="|5C^- "^^^^ 

Hence, by (2), <^ = - cT e'^J^ ^^^M^CT^) ^* ^^"^^ 

This gives, for x = 0, 

^ j = C I Jq (km) sin ia -jT- 4- CtD I Jq {km) sin kadk, . . (18) 

after a partial integration. The value of the former integral is given in (14), 
and that of the latter can be deduced from it by difEerentiation with respect 
to fD. Hence 

-^).-*'<'-»"'H-S-vi^:^) "" 

according as id ^ a. It follows that if C = 2/7r . Z7, the formula (17) will 
relate to the motion of a thin circular disk with velocity U normal to its 
plane, in an infinite mass of liquid. The expression for the kinetic energy is 

2T = - /> |[^ 1^ (iS = 7r/>C* ["* V(a* - in*) 27Tmdm = }7r*/>a»C*, 
or 2T = ipa^U* (20) 

The effective addition to the inertia of the disk is therefore 2/7r (= '6366) 
times the mass of a spherical portion of the fluid, of the same radius. For 
another investigation of this question, see Art. 108. 

* H. Weber, Crelle^ t. Ixxv. ; Part Diff,-Ql. t. i p. 189; Gray and Mathews, p. 126. See 
also Proc» Lond. Math. Soc. t. xxziv. p. 282. 



102-104] EUipsoidal Hamwnics 133 



EUipsaidal Harmonics. 

103. The method of Spherical Hannonics can also be adapted to the 
solution of the equation 

VV = 0, (1) 

under boundary-conditions having relation to ellipsoids of revolution*. 

Beginning with the case where the ellipsoids are prolate, we write 
x = k cos cosh Tj — kfi^y y = w cos co, 2 = to sin co,| 
where to = Asin ffsinhiy = ifc (1 - /i«)* (C« - 1)*. J 

The surfaces ^ = const., /x = const, are confocal ellipsoids, and hyperboloids 
of two sheets, respectively, the common foci being the points {± k, 0, 0). The 
value of ^ may range from 1 to oo , whilst /x lies between ± 1. The co-ordinates 
fif Ci <*> form an orthogonal system, and the values of the linear elements 
Stf^y is^, hs^ described by the point {x, y, z) when /i, {, co separately vary are 

^^ = * ir^) ^f"' ^'i' * (frzf ) ^^' 8»- = * (1 - /*•)* (i* - 1)* s*"- 

• \v) 

To express (1) in terms of our new variables we equate to zero the total 
flux across the walls of a volume element 8«^S«;S««, and obtain 

I (a|s.^.) V +1 (|s..8..) H + ^ (|8.A) 8» - 0, 

or, on substitution from (3), 
This may also be written 

dfi 0^ '^ ^a/ij ^ 1 - /i« aco» 35 r ^ ^ acj ^ 1 - c« aco«' • -^^^ 

104. If ^ be a finite function of /a and co, from /a=: — lto/A = + l and 
from CO = to CO » in, it may be expanded in a series of surface harmonics of 
integral orders, of the types given by Art. 86 (7), where the coefficients are 
functions of £ ; and it appears on substitution in (4) that each term of the 
expansion must satisfy the equation separately. Taking first the case of the 
zonal harmonic, we write 

^-P,(/i).Z, (5) 

* Heine^ "Ueber einige Aufgaben, welohe auf paitielle DifferentialgleiohimgeEi fahren," 
OreUe, t. xxyi p. 185 (1843), and Kugdfunitionen, t. iL Art. 38. See also Ferrers, o. rl 



134 Irrotational Motion of a Liquid [chap, v 

and on substitution we find, in virtue of Art. 84 (1), 

^j i<^ - ^*4|! + " ^" + ^) ^ "^ °' (^) 

which is of the same form as the equation referred to. We thus obtain the 

solutions 

^ - P, (/i) . P, (0 (7) 

and ^ = P, (,*) . Q, (a (8) 

where 



Qn a) = Pn a) f 



f{i*n(C)}* ({*-!)' 



= iP, (0 log |i-J - ^^ P„_x (0 - ^^-P,_.(0-. . . . 



ni 



f 



ff,.^. . (n+l)(n + 2) 
1^ ^ 2(2n + 3) ^ 



1 . 3 . . . (2n + 1) 

(n+ 1) (n + 2) (n + 3) (n + 4) | 

^ 2.4(2nV3)(2n + 5) " ^ +...|. .-W 

The solution (7) is finite when ^ = 1, and is therefore adapted to the 
space within an ellipsoid of revolution; whilst (8) is infinite for ^ = 1, but 
vanishes for £ = qo , and is therefore appropriate to the external region. As 
particular cases of the formula (9) we note 

e.(0 = iiog|^. ei(0 = inog[^J-i. 

Q» (0 = i (3C* - 1) log 1^ - K- 

The definite-integral form of Q„ shews that 

The corresponding expressions for the stream-function are readily found ; 
thus, from the definition of Art. 94, 



(11) 



whence |= - *(C* " D |f . | = M1-/^«)| (12) 

Thus, in the case of (7), we have 



<^-'^-¥-'-^i<'-'">'^'}' 



n (n + 1) 

* Fenren, c. t. ; Todhunter, o. vi. ; Fonyth, Arts. 9S-99. 



104r-106] Motion of an Ovary Ellipsoid 136 

"^«°«« ^ = M^) ^' - "'^'^^^ • ^^' - '^ ^ ^''^ 

The same result will follow of course from the second of equations (12). 
In the same way, the stream-function corresponding to (8) is 

105. We can apply this to the case of an ovary ellipsoid moving parallel 
to its axis in an infinite mass of liquid. The elUptic co-ordinates must be 
chosen so that the ellipsoid in question is a member of the confocal family, 
say that for which ^ = Co* Comparing with Art. 103 (2) we see that if a, c 
be the polar and equatorial radii, and e the eccentricity of the meridian 
section, we must have 

The surface-condition is given by Art. 97 (1), viz. we must have 

- - \Uk^ (1 - /x«) (C« - 1) + const., (1) 

for C=» Co- Hence putting n= 1 in Art. 104 (14), and introducing an 
arbitrary multiplier -4, we have 

^ = i^i(l-M»)({*-l)|ilog|3-}-^r^}, (2) 

with the condition 



(9) 



The corresponding formula for the velocity-potential is 

^ = ^M{Klog|^-l[ (4) 

The kinetic energy, and thence the inertia-coefficient due to the fluid, 
may be readily calculated, if required, by the formula (5) of Art. 94. 

106. Leaving the case of symmetry, the solutions of V*^ = when 
^ is a tesseral or sectorial harmonic in /a and a> are found by a similar 
method to be of the types 

^ = P,'(/*)..P«*U)3^)««, (1) 

^ = P„' (f*) . e»' (0 3°^} «-. (2) 

where, as in Art. 86, P,*(/x) = (1 - ^*)** '^'^"f^^ (3) 



136 



Irrotational Motion of a Liquid 



[chap. V 



whilst (to avoid imaginaries) we write 

and g„.(i;) = ((;._i)i.*|«l^). 

It may be shewn that 



(4) 



(5) 



QO 



dt 



whence P,« (0 ^%^ - ^^^ Qn' (0 = (-)'+^ 



{ {Pn' («}* . (C« - 1) ' • ■ 

(n + ») I 1 



di 



dC 



(n-»)!i»-r 



..(6) 



..(7) 



As examples we may take the case of an ovary ellipsoid moving parallel 
to an equatorial axis, say that of y, or rotating about this axis. 

1°. In the former case, the surface-condition is 
ioT ^ — ^, where F is the velocity of translation, or 



?^=_ 



F. 



K 



- ..t^i 



«-— , . — ; ^ (1 — U*)' 008 ft» 

This is satisfied by putting n = 1, « = 1, in (2), viz. 

^ = 4 (1 - M*)* (i«- 1)* . {i log 1^ - -^Fin} «» *-' 

the constant A being given by 



(8) 



...(9) 



^fi, ^0 + 1 ^o*-2 ) 



= -kV. 



(10) 



2°. In the case of rotation about Oy, if fly be the angular velocity, we 
must have 



for f = Jo or ^ = k*n,y . j . M (1 — M*)*8in co. 

35 (Co* - 1)* 

Putting n = 2, « « 1, in the formula (2) we find 



(11) 



^ = ^/.(l~ ,.«)»(£«- l)*|fClog|^- 3 -^^jsin CO, ..(12) 



A being determined by comparison with (11). 



106-107] Formulae far Planetary EUipBoid 137 

107. When the ellipsoid is of the (Mode or 'planetary' form, the 
appropriate co-ordinates are given by 

a? s=r i cos sinh ly = i/xf, y = to cos co, z = m sin (a,^ ., . 

it V • • . . ^1 j 

^««*« «/ — n,«x**^ v^*x.; — ,..VX — p y ( J* + l)^ J 

Here £ may range from to oo (or, in some applications from — oo through 
to + 00 ), whilst /A Ues between db 1. The quadrics ^ «= const., /i =» const, 
are planetary ellipsoids, and hyperboloids of revolution of one sheet, all 
having the common focal circle x = 0, g7 = A;. As Umiting forms we have the 
ellipsoid ^ = 0, which coincides with the portion of the plane a; == for which 
w <hy and the hyperboloid /a = coinciding with the remaining portion of 
this plane. 

With the same notation as before we find 

8», = *(f-±^^,*) 8,., 8«f = A (^l±-^y 8f , 8«. = i(l-/*»)»(i:«+l)*8«,; 

(2) 

and the equation of continuity becomes 

This is of the same form as Art. 103 (4), with t^ in place of C> <^nd the like 
correspondence will run through the subsequent formulae. 

In the case of sjrmmetry about the axis we have the solutions 

^ = p, (,*) . p, (a w 

and ^ = ^« (f*) . 3« (0. (5) 



. ,y. 1. 3.5 ... (2n- 1) 
where j)„ (0 = -j^ 



^ ^ 2 (2n - 1) ^ 



n(n-l)(n-2)(n-3) 1 

^ 2.4(2n-l)(2n-3) ^ -l-.-.J, --W 

and . gn(0-y,(0/; ^^(^)^f[^,^i) . 

= ♦>' f r-,-1 _ (n+l)(n + 2) , 

1.3.5 ... (2n+l) r 2(2n + 3) ^ 

(n + 1) (n + 2) (n + 3) (n + 4 ) .__, _ ) ,7. 

"^ 2.4(2n + 3)(2n + 6) ^ '**J 



138 IrrotdtioncU' Motion of a Liquid [ohap. v 

the latter expansion being however convergent only when J > 1 *. As before, 
the solution (4) is appropriate to the region included within an ellipsoid of 
the family ^ = const., and (5) to the external space. 

Wenotethat p„ (0^^^ - ^^^^,(0 = - ^^^ (8) 

As particular cases of the formula (7) we have 

Jo (C) = cot-i t, ?i (C) = 1 - { cot-» C, 

?.(a-i(3i*+l)cot-U-fC. 

The formulae for the stream-function corresponding to (4) and (6) are 

*-»-isTi)"-''''^'<^+'>%r^ '«' 

108. V. The simplest case of Art. 107 (5) is when n = 0, viz. 

^ = 24cot-iJ, (1) 

where ^ is supposed to range from — qo to + <3o . The formula (10) of the 
last Art. then assumes an indeterminate form, but we find by the method 
of Art. 104, 

^^Akfi (2) 

This solution represents the flow of a liquid through a circular aperture in 
an infinite plane wall, viz. the aperture is the portion of the plane yz for 
which m <h. The velocity at any point of the aperture (J = 0) is 

IdJf A 

since, when a? = 0, i/x = (i* — tn*)*. The velocity is therefore infinite at the 
edge. Compare Art. 102, 3°. 

2°. Again, the motion due to a planetary ellipsoid (J = Jo) moving with 
velocity TJ parallel to its axis in an infinite mass of Uquid is given by 

^ = ^/i(l-Jcot-U), 0-i^*(l-/*»)(5*+l)j^-cot-n},...(3) 

where A = — hU -h- [tt^^ — cot"^ Jo[ • 

Denoting the polar and equatorial radii by a and c, and the eccentricity 
of the meridian section by 6, we have 

a = iJo, c = i (U + 1)*, e = (V + 1)-*. 

* The reader may easily adapt the demonstrations referred to in Art. 104 to the present case. 



107-108] Streaan-Idnes of a Circular Disk 

In terms of these quantities 

il-- J7c4-|(l-e»)*-j8m-icl. . 



139 



...(4) 



The forms of the lines of motion, for equidistant values of 0, are shewn 
below. Cf. Art. 71, 3°. 




X' 



X 




The most interesting case is that of the circular disk, for which 6 = 1, 
and A ^ 2Uc/Tr. The value of <f> given in (3) becomes equal to ± Afi, or 

± -4 (1 — m^jt^y, for the two sides of the disk, and the normal velocity 
to ± U. Hence the formula (4) of Art. 44 gives 

2T^^p<^U\ (5) 

as in Art. 102 (20). 



140 Irrotcttional Motion of a Liquid [chap, y 

109. The solutions of the equation Art. 107 (3) in tessecal harmonics 



are 



and 



^ - P.* (^) . p/ (0 . ^1 to, (1) 

^ = P„* (/t) . ?«• (C) . ^} to, (2) 



where j,,.(^) = (^« + l)f^lO, (3) 



and g,.(0 = (C»+l)*'^^, 
These functions possess the property 

^ . a) ^n a) _ dp^'io / y+i(!L±iL' J_ (5) 

^•^ ^^^ dC dC *•* ^^' ^ ' (n-*)!C«+r •••^^ 

We may apply these results as in Art. 108. 

1^. For the motion of a planetary ellipsoid (^ = £o) parallel to' the axis 
of y we have n = 1, « = 1, and thence 

^-4(l-,.«)*(J«+l)*|^^-cot-icJcosa>, (6) 

with the condition ^ — — ^ ^> 



for J = ^o> y denoting the velocity of the solid. This gives 



^ld^-^°*"4=-*'^- 



(7) 



In the case of the disk (^q <= 0), we have ^ » 0, as we should expect. 



2°. A^in, for a planetary eUipeoid rotating about the axis of y with 
angular velocity Oy, we have, putting n = 2, « = 1, 

^ = ^/* (1 - m*)* a* + 1)* JSC cot-1 i - 3 + ^^1 sin «, . . .(8) 
with the surface-condition 

= *!0» . u (1 - u«)* sin a> (9) 

iU + 1)* 



109-110] EUipsoidal Envelope 141 

For the circular disk (f^ »= 0) this gives 

%7tA = - hm^ (10) 

At the two surfaces of the disk we have 

4>^^2Ayi(l^ /i«)*sinco, 1^ = T Wl^ (1 - /i«)*sinco, 
-and substituting in the formula 

we obtain 2T = ^pc^ . il^* (11) 

110. In questions relating to eUipeoids with three unequal axes we may 
employ the more general type of ElUpeoidal Harmonics, usually known by the 
name of * Lamp's Functions*.' Without attempting a formal account of these 
functions, we will investigate some solutions of the equation 

VV-=0, (1) 

in ellipsoidal co-ordinates, which are analogous to spherical harmonics of the 
first and second orders, with a view to their hydrodynamical appUcations. 

It is convenient to prefix an investigation of the motion of a liquid 
contained in an eUipeoidal envelope, which can be treated at once by 
Cartesian methods. 

Thus, when the envelope is in motion parallel to the axis of x with 
velocity U, the enclosed fluid moves as a solid, and the velocity-potential is 
simply <^ == — Ux. 

Next let us suppose that the envelope is rotating about a principal axis 
(say that of x) with angular velocity Q^^. The equation of the surface being 

X* V* z^ 

a* 6* c* ^ ' 



the surface-condition is 

a* dx 6« ay c« dz " 6« *^ ^ c« ^•^• 

We therefore assume <f> = Ayz, which is evidently a solution of (1), and 
obtain, on determining the constant by the condition just written, 

6* — c* 

* See, for example, Ferrers, Spherical Harmonics, o. vL; W. D. Niven, PhiL Trans. A, 
t. clzxxii. (1801) and Proc, Roy. Soc. A, t. Ixxix. p. 458 (1906); Poincar^, Figures (T^quilibre 
d^une Masse Fluide, Paris, 1902, c. vi. ; Darwin, PhU Trans. A, t. cxovii p. 461 (1901) [ScienHfic 
Papers, Cambridge, 1907-11, t. ill. p. 186]. An outline of the theory is given by Wangerin, 
Lc. ante p. 103. 



142 Irrotational Motion of a Liquid [chap, v 

Hence, if the centre be moving with a velocity whose components are 
TJy F, W and if !!«, fl^, fl, be the angular velocities about the principal axes, 
we have by superposition* 

4> Vx-Vy-Wz-^^^n,yz-'p~n^zx-^^il,xy....{Z) 

We may also include the case where the envelope is changing its form 
as well as position, but so as to remain elUpsoidal. If the axes are changing 
at the rates d, 6, c, respectively, the general boundary-condition, Art. 10 (3), 
becomes 

g* + ^'^3* + Sl + ^| + S|-»- («) 

which is satisfied t by 

The equation (1) requires that 

^ + - + !: = 0, 6) 

a b c 

which is in fact the condition which must be satisfied by the changing 
ellipsoidal surface in order that the enclosed volume (^abc) may be constant. 

111. The solutions of the corresponding problems for an infinite mass 
of fluid bounded internally by an ellipsoid involve the use of a special system 
of orthogonal curvilinear co-ordinates. 

If X, y, z be functions of three parameters A, /a, v, such that the surfaces 

A = const., fi = const., v = const (1) 

are mutually orthogonal at their intersections, and if we write 



l,.(8£)V(|!y+(|!)', 

V \9/^/ V3ft/ \dfiJ 

»!.-©'-g)'-©'-J 



(2) 



* This result appears to have been published independently by Beltrami, Bjerknes, and 
Maxwell, in 1873. See Hicks, "Report on Recent Progress in Hydrodynamics," BriL Asa. Rep. 
1882, and Kelvin's Papers, t. iv. p. 197 (footnote). 

t C. A. Bjerknes, "Verallgemeinerung des Problems von den Bewegungen, welohe in einer 
ruhenden unelastischen FlUssigkeit die Bewpgung eines Ellipsoids hervorbringt," OWinger 
Nachrichten, 1873. 



110-112] Orthogonal Co-ordinates 143 

the direction-cosines of the normals to the three surfaces which pass through 
(», y, z) will be 

paA' '^aX' *^aAJ' V^v *«v **8^j' l^'a;;' *»ai;' *»ci;J' •••<3) 

respectively. It follows that the lengths of linear elements drawn in the 
directions of these normals will be 

SA/Ai, Sft/Aj} Sv/Ag. 

Hence if <^ be the velocity-potential of any fluid motion, the total flux 
into the rectangular space included between the six surfaces A db |8A, ^db J8^, 
v± \hv will be 

It appears from Art. 42 (3) that the same flux is expressed by ^^<f> multiplied 
by the volume of the space, i.e. by SXhfiSv/hih^h^, Hence* 

^■*-»A».g(j^J)^|.(i|)4(4|*))...(4) 

Equating this to zero, we obtain the general equation of continuity in 
orthogonal co-ordinates, of which particular cases have already been investi- 
gated in Arts. 83, 103, 108. 

The theory of triple orthogonal systems of surfaces is very attractive 
mathematically, and abounds in interesting and elegant formulae. We may 
note that if A, ft, v be regarded as fimctions of x, y, z, the direction-cosines of 
the three line-elements above considered can also be expressed in the forms 

[h^dx' h^dy' h^dzj' [h^dx' h^dy' h^dzj' [h^dx' h^dy' h^dzj' 

(5) 

from which, and from (3), various interesting relations can be inferred. The 
formulae already given are, however, sufficient for our present purpose. 

112. In the applications to which we now proceed the triple orthogonal 
system consists of the conf ocal quadrics 

• "* +r/\+.-^.-l = 0. (1) 



a^-\-e b^ + e c^ + d 

* The above method was given in a paper by W. Thomson, *'0n the Equations of Motion of 
Heat referred to Curvilinear Co-ordinates/' Camb. Math. Joum. t. iv. (1843) [Papers, t. i. p. 25]. 
Reference may also be made to Jacobi, "Ueber ^ine particul&re Losung der partiellen Diffe- 
rentialgleiohung ," CreUe, t. xxxvi. (1847) [Werke, t. ii. p. 198]. 

The transformation of V*0 to general orthogonal co-ordinates was first effected by Lam^, '*Sur 
les lois de T^quilibre du fluide ^th^r6," Joum. de T£c6U Polyi, t. ziv. (1834). See also Lefons 
9ur les Coordonndea Curvilignee, Paris, 1869, p. 22. 



144 



Irrotational Motion of a Idquid 



[OHAP. V 



whose properties are explained in books on Solid Geometry. Through any 
given point (x, y, z) there pass three surfaces of the system, corresponding 
to the three roots of (1), considered as a cubic in 0. If (as we shall for the 
most part suppose) a > 6 > c, one of these roots (A, say) will lie between qo 
and — c\ another (/x) between — c* and — 6*, and the third (v) between — 6* 
and — a*. The surfaces A, ft, v are therefore ellipsoids, hyperboloids of one 
sheet, and hyperboloids of two sheets, respectively. 

It follows immediately from this definition of A, /a, v, that 



x^ 



+ 



.7* 



+ 



_^ (\^d)(ii^d)(v-d) 

5« + (a« + e) (6« + 0) (c« + <?)'•• • '^^^ 



a« + d ^ 6« + c 

identically, for all values of 6. Hence multiplying by a* + d, and afterwards 
putting = — a^y we obtain the first of the following equations : 



x^ 



y' 



««« 



(a« + A) ( g g + fi) (g' + v) 
(g« - 6«) (g2 - c2) 

(6« + A) (6« + ji) (6« + i;) 
(6* - c«) (6« - g«) 

(c« + A) (c« 4- /x) (c2 + v) 



(c« - g«) (c« - 6«) 



(3) 



These give 



Sx _ . X 



^_ 



i 



y 



a»+A' aA 2 6« + A' aA~'3*TA' 



92 _ J z 



(4) 



and thence, in the notation of Art. Ill (2), 






+ 



y* 



:.+ 



A)* ^ {b* + A)» "^ (c» + A)«J 



(5) 



If we differentiate (2) with respect to 6 and afterwards put fl == A, we deduce 
the first of the following three relations : 



i.._. («' + ^)(ft* + A)(c« + A) ] 
(A-^)(A-v) • 

A « = 4 («* + m) (ft* + m) (c* + li) 

(m - v) (/*'- A) ' 

;i 2 = 4 (o' + v) (6* + v) {c* +v) 
* (v-A)(v-/t) • 



(6) 



The remaining relations of the sets (3) and (6) have been written down from 
symmetry*. 



• It will be noticed that *j. A,, A, are double the perpendioolan from the origin on the 
tangent planes to the three quadrios X, n, r. 



112-113] Orthogonal Co-ordinates 145 

Substituting in Art. Ill (4), we find* 

+ (v - A) |(a« + ,*)* (6« + ^)» (c« + ,*)» 1^1* 

+ (A - /*) |(o« + v)* (6* + v)* (c» + v)* l^ri 4>. 

(7) 

113. The particular solutions of the transformed equation V^ = which 
first present themselves are those in which <^ is a function of one (only) of 
the variables A, /a, v. Thus ^ may be a function of A alone, provided 

(a« + A)* (62 + A)* (c* + A)*^ = const., 

whence <f> = C j -r-, (1) 

if A = {(a« + A) (6« + A) (c« + A)}*, (2) 

the additive constant which attaches to <f> being chosen so as to make <f> 
vanish for A = oo . 

In this solution, which corresponds to ^ = A/r in spherical harmonics, 
the equipotential surfaces are the confocal ellipsoids, and the motion in the 
space external to any one of these (say that for which A = 0) is that due to a 
certain arrangement of simple sources over it. The velocity at any point is 
given by the formula 

-».S-4 <») 

At a great distance from the origin the ellipsoids A become spheres of 
radius A*, and the velocity is therefore idtimately equal to 2C/r*, where r 
denotes the distance from the origin. Over any particular equipotential 
surface A, the velocity varies as the perpendicular from the centre on the 
tangent plane. 

To find the distribution of sources over the surface A = which would 
produce the actual motion in the external space, we substitute for <f> the 
value (1), in the formula (11) of Art. 58, and for <f)' (which refers to the 
internal space) the constant value 

•*dA 



^'-af w 



* Cf. Lam^, "Sur les surfaces itothermeB dans les corps solides homog^nes en ^quilibre de 
temperature," LiouviOe, t. u. (1837). 

L. H. 10 



146 Irrotational Motion of a Liquid [chap, v 

The formula referred to then gives, for the surface-density of the required 
distribution, 

ic-^' (^) 

The solution (1) may also be interpreted as representing the motion due 
to a change in the dimensions of the ellipsoid, such that the surface remains 
similar to itself, and retains the directions of its principal axes unchanged. 
If we put 

d/a = b/b = c/c, = A, say, 

the surface-condition Art. 110 (4) becomes 

— d<f>/dn = ^khi, 
which is identical with (3), if we put C = Ikdbc. 

A particular case of (5) is where the sources are distributed over the 
elliptic disk for which A = — c^, and therefore z* = 0. This is important in 
Electrostatics, but a more interesting application from the present point of 
view is to the flow through an elliptic aperture, viz. if the plane ay be 
occupied by a thin rigid partition with the exception of the part included by 
the ellipse 

^2 + p = i' « = 0' 

we have, putting c = in the previous formulae, 

(6) 



■/ (a* + A)* (6« + A)* A* ' 



where the upper limit is the positive root of 



""^ +./^^ + T = l, (7) 



a* 4- A 62 4- A A 

and the negative or the positive sign is to be taken according as the point 
for which <f> is required lies on the positive or the negative side of the plane 
a>y. The two values of <f> are continuous at the aperture, where A = 0. As 
before, the velocity at a great distance is equal to 2^/r*, nearly. For points 
in the aperture the velocity may be found immediately from (6) and (7) ; thus 
we may put 



Sn 



.....(x-g-g)', 8, = -!^* 



approximately, since A is small, whence 

This becomes infinite, as we should expect, at the edge. The particular case 
of a circular aperture has already been solved otherwise in Arts. 102, 108. 



113-114] Translation of an Ellipsoid 147 

114. We proceed to investigate the solution of V*^ = 0, finite at infinity, 
which corresponds, for the space external to the elUpsoid, to the solution 
<f> = X ioT the internal space. Following the analogy of spherical harmonics 
we may assume for trial 

<f> = ^y (1) 



which gives V^X + - 5^ = 0, 



xdx 



(2) 



and inquire whether this can be satisfied by making x equal to some function 
of A only. On this supposition we shall have, by Art. Ill, 

dx'^Hy^^dr 

and therefore, by Art. 112 (4), (6), 

xdx (A — /i) (A — v) dX' 
On substituting the value of V*;^ in terms of A, the equation (2) becomes 

|(a« + A)* (6« + A)» (c« + A)* ^\=- (6* + A) (c* + A) ^, 
which may be written 

^,log{(a3+A)*(63 + A)t(c^ + A)*|| = -^,. 

Hence X^oT . -, ., (3) 

J M«^ + A)* (6* + A)t (c« + A)* 

the arbitrary constant which presents itself in the second integration being 
chosen as before so as to make x vanish at infinity. 

The solution contained in (1) and (3) enables us to find the motion of a 
liquid, at rest at infinity, produced by the translation of a solid ellipsoid 
through it, parallel to a principal axis. The notation being as before, and 
the ellipsoid 

o« ^ 6« ^ c» 



+ Si + !i=l (4) 



being supposed in motion parallel to x with velocity U, the surface- 
condition is 

|=-17|. forA = (5) 

Let us write, for shortness, 

(6) 

10—2 



148 Irrotational Motion of a Liquid [chap, v 

where A = {(a^ + A) (6* + A) (c^ + A)}* ; (7) 

it will be noticed that these quantities <io, jSq, y© *re purely numerical. The 
conditions of our problem are satisfied by 

* = C./;p^. (8) 

provided C = ^— — U (9) 

The corresponding solution when the ellipsoid moves parallel to y or 2; 
can be written down from symmetry, and by superposition we derive the 
case where the ellipsoid has any motion of translation whatever*. 

At a great distance from the origin, the formula (8) becomes equivalent to 

<A = *c^, (10) 

which is the velocity-potential of a double source at the origin, of strength 
§7rO, or 

compare Art. 92. 

m 

The kinetic energy of the fluid is given by 

where I is the cosine of the angle which the normal to the surface makes 
with the axis of x. Since the latter integral is equal to the volume of the 
elUpsoid, we have 

2T = -^r^^— . ^nabcp . U^ (11) 

The inertia-coefl&cient is therefore equal to the fraction 0^/(2 — Oq) of the 
mass displaced by the soUd. For the case of the sphere (a = 6 = c) we find 
oo = |; this makes the fraction equal to i, in agreement with Art. 92. If 
we put 6 = c, we get the case of an ellipsoid of revolution, including (for a = 0) 
that of a circular disk. The identification with the results obtained by the 
methods of Arts. 105, 106, 108, 109 for these cases may be left to the reader. 

* This problem was first solved by Green, "Researches on the Vibration of Pendulams in 
Fluid Media " Trans. B. S. Edin. 1833 [Papers, p. 315]. The investigation is much shortened 
if we assume at once from the Theory of Attractions that (8) is a solution of V*0=O, being in 
fact (except for a constant factor) the :r-component of the attraction of a homogeneous ellipsoid 
at an external point. 



114-115] Rotation of an EUipaoid 149 

115. We next inquire whether the equation V*^ = can be satisfied by 

4> = y^Xy (1) 

where x is » function of A only. This requires 

^--^itAi-"- (2) 

Now, from Art. 112 (4), (6), 

ydy zdz ^ \y 3A z dX) dX 



_ (o« + A) (6« + A) (c' + A) / 1 1 \dx 

(A - u) (A - v) U* + A "*" c» + A/ dX' 



(A-/*){A-v) 
On substitutioQ in (2) we find, bj Ait. 112 (7), 

^, log {(«« + A)» (6. + A)» (c. + A)* 1} = - ^ - 3^. 

whence X° gj, (fc. + A) (o« + A) A ^^^ 

the second constant of integration being chosen as before. 

For a rigid ellipsoid rotating about the axis of x with angular velocity 
Qajj the surface-condition is 

I-°.(»|-»IS- •• <*) 

for A = 0. Assuming* 

dX 



<f> = Cyzf 



./* (6» + A) {c« + A) A ^^^ 

we find that the sniface-condition (4) is satisfied, provided 

The formulae for the cases of rotation about y ot z can be written down from 
symmetry f. 

* The expression (5) differs only by a factor from 

where 4 is the gravitation-potential of a nniform solid ellipsoid at an external point (a;, y^ 2;). 
Since V*<^=0 it easily follows that the above is also a solution of the equation VV=0. 

t The solution contained in (5) and (6) is due to Qebsch, "Ueber die Bewegung eines 
EOipsoidee in einer tropfbaren Flussigkeit," Crdit, tt. lii. liii. (1866-7). 



150 Irrotational Motion of a Liquid [chap, v 

The formula for the kinetic energy is 

if ({, m, n) denote the direction-cosines of the normal to the ellipsoid. The 
latter integral 

= Hi (y' - 2*) dxdydz = H&* - '«') • l^«^. 
Hence we find 

The two remaining types of ellipsoidal harmonic of the second order, finite at the origin, 
are given by the expression 

x" y* 2* 



a*+^ h^ + B c^-^B 



-1 (8) 



where B is either root of •- :;i + r* - >» + -? /» =0 (9) 

this being the condition that (8) should satisfy v*^ =0. 

The method of obtaining the corresponding solutions for the external space is explained 
in the treatise of Ferrers. These solutions would enable us to express the motion produced 
in a surrounding liquid by variations in the lengths of the axes of an ellipsoid, subject to 
the condition of no variation of volume : 

dja -^b/b +c/c=0 (10) 

We have already found, in Art. 113, the solution for the case where the eUipsoid expands 
(or contracts) remaining similar to itself; so that by superposition we could obtain the 
case of an internal boundary changing its position and dimensions in any manner what- 
ever, subject only to the condition of remaining eUipsoidal. This extension of the results 
arrived at by Green and Clebsch was first treated, though in a different manner from 
that here indicated, by Bjerknes*. 

116. The investigations of this chapter have related almost entirely 
to the case of spherical or ellipsoidal boundaries. It will be understood 
that solutions of the equation V^ = can be carried out, on lines more 
or less similar, which are appropriate to other forms of boundary. The 
surface which comes next in interest, from the point of view of the present 
subject, is that of the anchor-ring, or 'torus'; this case has been very ably 
treated, by distinct methods, by Hicks, and Dyson f. We may also refer to 
the analytically remarkable problem of the spherical bowl, which has been 
investigated by Basset J. 

* Le. ante p. 142. 

t Hicks, "On Toroidal Functiona," Phil. Trans. 1881; Dyaon, "On the Potential of an 
Anchor-Ring," PhU. Trans. 1893 ; see also 0. Neumann, Ic. ante p. 127. 

X "On the Potential of an Electrified Spherical Bowl, &c.," Proc. Lond. Math. 8oc. t. zvi. 
(1885); Hydrodynamics, t. i. p. 149. 



CHAPTER VI 

ON THE MOTION OF SOLIDS THROUGH A LIQUID : 

DYNAMICAL THEORY 

117. In this chapter it is proposed to study the very interesting 
dynamical problem furnished by the motion of one or more solids in a 
frictionless liquid. The development of this subject is due mainly to 
Thomson and Tait* and to Kirch hofff* 1*^® cardinal feature of the methods 
followed by these writers consists in this, that the solids and the fluid 
are treated as forming one dynamical system, and thus the troublesome 
calculation of the effect of the fluid pressures on the surfaces of the soUds 
is avoided. 

To begin with the case of a single solid moving through an infinite mass 
of liquid, we will suppose in the first instance that the motion of the fluid is 
entirely due to that of the solid, and is therefore irrotational and acyclic. 
Some special cases of this problem have been treated incidentally in the 
foregoing pages, and it appeared that the whole effect of the fluid might be 
represented by an addition to the inertia of the solid. The same result will 
be found to hold in general, provided we use the term 'inertia' in a somewhat 
extended sense. 

Under the circumstances supposed, the motion of the fluid is characterized 
by the existence of a single-valued velocity-potential ^ which, besides 
satisfying the equation of continuity 

vv = o, (1) 

fulfils the following conditions : (1°) the value of — 3^/3n, where 8n denotes 
as usual an element of the normal at any point of the surface of the solid, 
drawn on the side of the fluid, must be equal to the velocity of the surface 
at that point normal to itself, and (2°) the differential coefficients d<f>/dx, 

i 

* Natural Philosophy, Art. 320. Subsequent inveetigatioiiB by Lord Kelvin will be referred 
to later. 

t "Ueber die Bewegnng eines Rotationakorpers in einer Fliissigkeit/' CreUe, t. Ixzi. (1869) 
[Qea, AhK p. 376]; Mechanik, c. ziz. 



152 Motion of Solids through a Liquid [chap, vi 

d<f>ldy, d<f>ldz must vanish at an infinite distance, in every direction, from the 
solid. The latter condition is rendered necessary by the consideration that 
a finite velocity at infinity would imply an infinite kinetic energy, which 
could not be generated by finite forces acting for a finite time on the solid. 
It is also the condition to which we are led by supposing the fluid to be 
enclosed within a fixed vessel infinitely large and infinitely distant, all round, 
from the moving body. Foi.' on this supposition the space occupied by the 
fluid may be conceived as made up of tubes of flow which begin and end on 
the surface of the solid, so that the total flux across any area, finite or 
infinite, drawn in the fluid must be finite, and therefore the velocity at 
infinity zero. 

It has been shewn in Art. 41 that under the above conditions the motion 
of the fluid is determinate. 

118. In the further study of the problem it is convenient to follow the 
method introduced by Euler in the dynamics of rigid bodies, and to adopt a 
system of rectangular axes Ox, Oy, Oz fixed in the body, and moving with it. 
If the motion of the body at any instant be defined by the angular velocities 
p, J, r about, and the translational velocities w, v, w of the origin parallel 
to, the instantaneous positions of these axes*, we may write, after Kirchhoff, 

<l> = u<f>i + v<f>^-hw<f>^ + pXi + qXi + rxsy (2) 

where, as will appear immediately, ^i, ^2» ^3> Xi> Xa* Xs *^® certain functions 
of X, y, z determined solely by the configuration of the surface of the solid, 
relative to the co-ordinate axes. In fact, if I, m, n denote the direction-cosines 
of the normal, drawn towards the fluid, at any point of this surface, the 
kinematical surface-condition is 

■ 

— ^ = l{u -{- qz — ry) + m{v -{- rx — pz) + n{w -{- py — qx), 
whence, substituting the value (2) of <f>, we find 



dK -^' "a^T -^' "- =^ 



^^^^ny-mz. --^ = lz-nx, ^^=.mx^ly. 



\o) 



Since these functions must also satisfy (1), and have their derivatives zero at 
infinity, they are completely determinate, by Art. 41 f . 

* The symbols u, v, tc, p, q, r are not at present required in their former meanings, 
f For the particular ease of an ellipsoidal surface, their values may be written down from 
the results of Arts. 114, 116. 



117-119] ImpuUe of the Motion 153 

119. Now whatever the motion of the solid and fluid at any instant, it 
might have been generated instantaneously from rest by a properly adjusted 
impulsive 'wrench' applied to the soUd. This wrench is in fact that which 
would be required to counteract the impulsive pressures p<f> on the surface, 
and, in addition, to generate the actual momentum of the solid. It is called 
by Lord Kelvin the 'impulse' of the system at the moment under con- 
sideration. It is to be noted that the impulse, as thus defined, cannot be 
asserted to be equivalent to the total momentum of the system, which is 
indeed in the present problem indeterminate*. We proceed to shew 
however that the impulse varies, in consequence of extraneous forces acting 
on the solid, in exactly the same way as the momentum of a finite dynamical 
system. 

Let us in the first instance consider any actual motion of a solid, from 
time to to time ^^9 under any given forces applied to it, in a jmite mass 
of liquid enclosed by a fixed envelope of any form. Let us imagine the 
motion to have been generated from rest, previously to the time ^, by forces 
(whether continuous or impulsive) applied to the solid, and to be arrested, in 
like manner, by forces applied to the solid after the time t^. Since the 
momentum of the system is null both at the beginning and at the end of this 
process, the time-integrals of the forces appUed to the solid, together with 
the time-integral of the pressures exerted on the fluid by the envelope, must 
form an equilibrating system. The effect of these latter pressures may be 
calculated, by Art. 20, from the formula 

f = |-k' + y(0 (1) 

A pressure uniform over the envelope has no resultant effect ; hence, since <f> 
is constant at the beginning and end, the only effective part of the integral 
pressure Ipdtis given by the term 

-\pii^^ (2) 

Let us now revert to the original form of our problem, and suppose the 
containing envelope to be infinitely large, and infinitely distant in every 
direction from the moving solid. It is easily seen by considering the 
arrangement of the tubes of flow (Art. 36) that the fluid velocity j at a great 
distance r from an origin in the neighbourhood of the solid will ultimately 
be, at mostf, of the order 1/r*, and the integral pressure (2) therefore of the 
order 1/r*. Since the surface-elements of the envelope are of the order r^Stu, 
where hm is an elementary sohd angle, the force- and couple-resultants of 
the integral pressure (2) will now both be null. The same statement 

* That is, the attempt to calculate it leads to 'improper* or 'indeterminate' integralB. 

t It is really of the order l/r* when, as in the case considered, the total flux outwards is zero. 



154 Motion of Solids through a Liquid [chap, vi 

therefore holds with regard to the time-integral of the forces applied to 
the solid. 

If we imagine the motion to have been started insUvnianeously at time 
^0, and to be arrested instantaneously at time ^i, the result at which we have 
arrived may be stated as follows : 

The 'impulse' of the motion (in Lord Kelvin's sense) at time i^ differs 
from the 'impulse' at time ^o l>y the time-integral of the extraneous forces 
acting on the solid during the interval i^ — ^q*. 

It will be noticed that the above reasoning is substantially unaltered 
when the single solid is replaced by a group of soUds, which may moreover 
be flexible instead of rigid, and even when these solids are replaced by 
masses of Uquid which are moving rotationally. 

120. To express the above result analytically, let £, ly, J, A, /i, v be the 
components of the force- and couple-constituents of the impulse; and let 
X^ y, Z, i, Jf , iV designate in the same manner the system of extraneous 
forces. The whole variation of |, ry, ^, A, /i, i/, due partly to the motion of the 
axes to which these quantities are referred, and partly to the action of the 
extraneous forces, is then given by the formulae f 

= uyq ^ vQ + rfjL - qi^ -{- Ly 
^=u^--wi + pu^rX + M,\ (1) 

— = V^ - 7/7^ -f. grA - ;?/i 4- N. 

For at time t ■■{- &t the moving axes make with their positions at time t 
angles whose cosines are 

(1, rSt, - qSt), (- rSt, 1, pSt), (gr&, - p8t, 1), 

respectively. Hence, resolving parallel to the new position of the axis of x, 

^ + Si=i + rj.rSt-^.q8t + XBL 

Again, taking moments about the new position of Ox, and remembering 
that has been displaced through spaces uSt, vSt, w8t parallel to the 
axes, we find 

A + 8A = A + t; . t^8« - ^ . vSt + /i . r8« - V . gr8« + LSt. 

These, with the similar results which can be written down from symmetry, 
give the equations (1). 

♦ Sir W. Thomson, l.c. anie p. 31. The form of the argument given above was kindly 
suggested to the author by Sir J. Larmor. 

t Cf. Hayward, "On a Direct Method of Estimating Velocities, Accelerations, and all 
similai Quantities, with respect to Axes moveable in any manner in space,*' Camb, Trana. 
t. X. (1866). 



di' 


= r.j- 


-qUX, 


dri 
di' 


-vl- 


-ri+Y, 


dt 
dt' 


= qi- 


-pq+Z, 



119-121] 



Kinetic Energy 



155 



When no extraneous forces act, we verify at once that these equations 
have the integrals 

f* + 17* + i* = const., X^ -\- ii-q -\- vl = const., (2) 

which express that the magnitudes of the force- and couple-resultants of the 
impulse are constant. 

121. It remains to express |, -q, J, A, /i, 1/ in terms of w, v, w^ p, j, f. In 
the first place let T denote the kinetic energy of th^fiutd, so that 



2T 



—Mti^. 



m 



where the integration extends over the surface of the moving solid. 
Substituting the value of <f> from Art. 118 (2), we get 

2T = Aw« + Bv« + Gw^ + 2A't;u? + 2B'wu + 2G'uv 

+ Py« + Qj« + Rr* + 2P'3T + 2QVp + 2R'j)g 

+ 2p (Pw + Ov + Hw) + 2q (T'u + G'v + K'w) + 2r (P''w + Q'^v + B."w), 

(2) 

where the 21 coefficients A, B, C, &c. are certain constants determined by 
the form and position of the surface relative to the co-ordinate axes. Thus, 
for example, 

* 

— ,jj^^-iS.-,IJ^,^'^ \ (3) 



= " '' //^» fe' '^ = '' Ih' ("^ ~ "'''^ '^^• 



/ 



the transformations depending on Art. 118 (3) and on a particular case of 
Green's Theorem (Art. 44 (2)). These expressions for the coefficients were 
given by Elirchhoff. 



The actual values of the coefficients in the expression for 2T have been found in the 
preceding chapter for the case of the ellipsoid, viz. we have from Arts. 114, 116 



A— 5L. Uaabc p 1 (fe*-C)' ( y.-g.) 



. ^irpabc. 



(*) 



156 Motion of Solids through a Liquid [chap, vi 

with similar expressions for B, 0, Q, B. The remaining coefficients, as will appear pre- 
sently, in this case all vanish. We note that 

^-»=(2^Sy#k)-*''^' <') 

80 that if a > 6 > c, then A < B < C, as might have been anticipated. 

The formulae for an ellipsoid of revolution may be deduced by putting 6 =c ; they may 
also be obtained independently by the method of Arts. 104-109. Thus for a circular disk 
(a =0, 6 =c) we have 

A,B, C=ipc»,0,0; P,Q, B=0, H/'Cfi.ifpc*. (6) 

The kinetic energy, .T^ say, of the solid alone is given by an expression of 
the form 

2Ti =ft m (fi« -f t;2 -f w^) 

+ PiP" + Qi?" + Rir* + 2PiV + 2Qi'rp + 2Ri'^ 

+ 2in{a {vr — wq) + jS {wp — ur) -{■ y {uq — vp)} (7) 

Hence the total energy T + T, , of the system, which we shall denote by T, is 
given by an expression of the same general form as (2), say 

2r = Au^ + Bv^ + Cw^ + 2A'vw + 2B'wu + 2C'uv 

-f Pf + Qq^ + JRr" + 2P'gr + 2QWp + 2Rjq 

-V2p{Fu-\-Gv + Hw) + 2gr {F'u + G'v -f H'w) + 2r {F''u + G"v + H"w), 

_ (8) 

where the coefficients are printed in uniform type, although six of them have 
of course the same values as in (2) 

122. The values of the several components of the impulse in terms of 
the velocities u, v, w, p, q, r can now be found by a well-known dynamical 
method*. Let a system of indefinitely great forces (X, Y, Z, i, Jf, N) act 
for an indefinitely short time r on the soUd, so as to change the impulse from 
(L V> J» A, /i, v) to (f + 8|, ly + &7, J + S^ A + SA, /* + S/i, V + Si/). The work 
done by the force X, viz. 



/, 



Xudly 



Ues between Ui\ Xdt and w, I Xdt, 

Jo Jo 

where Ui and tig ^^^ ^^^ greatest and least values of u during the time r, 
i.e. it lies between w^S^ and WgSf . If we now introduce the supposition that 
8|, S?y, S^, SA, 8/Lt, 8v are infinitely small, Wj and w, are each equal to u, and 
the work done is wSf . In the same way we may calculate the work done by 

* See Thomson and Tait, Art-. 313, or Maxwell, Electricity and Magnetism, Part. iv. c. v. 



121-122] Relations betweefi Energy and Impvlse 157 

the remaming forces and couples. The total result must be equal to the 
increment of the kinetic energy, whence 

w8f + vSry + t^SJ + p8A + 58/x + rSv 

Now if the velocities be all altered in any given ratio, the impulses wiU 
be altered in the same ratio. If then we take 

8w _ Sv __ Si^ _ 8p _ 8} _ 8r _ , 
u ^ V " w ~ p q r ' 

it WiU follow that ^ = h = % ^^4 = ^-^ = - - ^■ 

Substituting in (1), we find 
ui + vr) + w^ + pX+ qfi + rv 

dT , dT , ar , dT ^ ar , ar „„ ,.,, 
= "a^ + ''a^ + «'a^ + Pa^ + «W"^''3^° ^' ••"^^^ 

since T is a homogeneous quadratic function. Now performing the arbitrary 
variation 8 on the first and last members of (2), and omitting terms which 
cancel by (1), we find 

Since the variations 8u, 8t;, iw^ 8p, Sq, 8r are all independent, this gives the 
required formulae 

^' ''^ ^ "" du' dv' dw' ^' ^'"''^ dp' dq' dr ^"^^ 

It may be noted that since f , t?, J, ... are linear functions of w, v, «?, . . . , 
the latter quantities may also be expressed as linear fimctions of the former, 
and thence T may be regarded as a homogeneous quadratic function of 
£> Vy i> ^> /*> *'• When expressed in this manner we may denote it by T\ 
The equation (1) then gives at once 

u8f + vS?y + wS^ + pSX 4- qSfjL 4- rSv 

= -af^^ + ^^ + -ar^^ + 'aA^^+a^^'* + -air^''' 
, ar dr dr dr ar dr 

whence u, v, w = .^, -^. j^ , p, q, r = -^j, ^-, ^ (4) 

These formulae are in a sense reciprocal to (3). 

We can utilize this last result to obtain, when no extraneous forces act^ 



158 



Motion of Solids through a Liquid [chap, vi 



another integral of the equations of motion, in addition to those found in 
Art. 120. Thus 

dt 'di (ft "^ • • • "^ • • • + dX (ft "^ • • • "^ • • • 

_ df ^ ^ dX^ 

-t^^-t- ... + ... +?^+ ... + ..., 

which vanishes identically, by Art. 120 (1). Hence we have the equation 

of energy 

T = const (5) 

128. If in the formulae (3) we put, in the notation of Art. 121, 

T = T + Ti, 
it is known from the dynamics of rigid bodies that the terms in T^ represent 
the linear and angular momentum of the solid by itself. Hence the remaining 
terms, involving T, must represent the system of impulsive pressures exerted 
by the surface of the soUd on the fluid, in the supposed instantaneous 
generation of the motion from rest. 

This is easily verified. For example, the x-component of the above 
system of impulsive pressures is 

= At^ + O'v + B*w + Pp + T'q -f P'V = gi, ... .(6) 

by the formulae of Arts. 118, 121. In the same way, the moment of the 
impulsive pressures about Ox is 

JJp^ (ny-mz)dS=-p fj<f> ^ dS 

3T 

=^Fu-\-aV'{-B.w + 'Pp+Ii'q-i-Q'r= '^ (7) 



dp' 



124. The equations of motion may now be written* 

d_dT^^dT_ 92; 
dtdu dv ^ dw ' 

dt dv ^dw du ' 



\ 



d dT dT 



dT 



ji* a-« "" ? a-. P Oi, "^ ^» 



(ft dw 



du 



dv 



dar_ dT^^^dT^ ^dT _ 92; 

dt dp "" dv dw dq ^ dr ' 

d dT dT dT , dT dT ^ ,, 



> 



(1) 



dt dq 
d dT 



dw 



du 
dT 



dr 



dp 



^ dT _ _ 

dtdr du dv 



^ dT 32"^ V f 



* See Kirchhoff, le. ante p. 151 ; also Sir W. Thomson, "Hydrokinetio Solations and Obser- 
vations," PhU, Mag. Nov. 1871 [reprinted in BaUimare Lectures, Cambridge, 1904, p. 684]. 



122-124] Equations of Motion 159 

If in these we write T = T + Ti, and isolate the terms due to T, 
we obtain expressions for the forces exerted on the moving solid by the 
pressure of the surroimding fluid ; thus the total component (X, say) of the 
fluid pressure parallel to x is 

and the moment (L) of the same pressures about cr is* 

L - - - — -I- 9T _ ar aT _ 8T 

dtdp dv dw dq ^ dr 

For example, if the solid be constrained to move with a constant velocity 
{u, V, w), without rotation, we have 



X, Y, Z = 0, 

~ ~ ~ \ 



T *, ^T 3T aT aT aT bt bt >....(4) 



dv dw* dw du* du dv* 

where 2T = Au^ + Bv* + Cw^ + 2A'vw + 2B'wu + 2C'uv. 

The fluid pressures thus reduce to a couple, which moreover vanishes if 

aT aT aT 

du dv dw 

i.e. provided the velocity (w, v, w) be in the direction of one of the principal 
axes of the eUipsoid 

Ax^ + By* + Cz^ + 2A'yz + 2B'zx + 2C'xy = const (5) 

Hence, as was first pointed out by Kirchhoff, there are, for any solid, 
three mutually perpendicular directions of permanent translation; that is 
to say, if the soUd be set in motion parallel to one of these, without 
rotation, and left to itself, it will continue to move in this manner. It 
is evident that these directions are determined solely by the configuration 
of the surface of the body. It must be observed however that the impulse 
necessary to produce one of these permanent translations does not in general 
reduce to a single force; thus if the axes of co-ordinates be chosen, for 
simplicity, parallel to the three directions in question, so that A\ JB', C = 0, 
we have, corresponding to the motion u alone, 

i, 7), J = Au, 0, 0; A, /i, 1/ == Fu, F% F"u, 

80 that the impulse consists of a wrench of pitch FjA. 

* The forms of these expressions being known, it is not difficult to verify them by direct 
calculation from the pressure-equation. Art. 20 (4). See a paper " On the Forces experienced by 
a Solid moving through a liquid,** Quart. Jowm. MtUk. t. xix. (1883). 



160 Motion of Solids through a Liquid [chap, vi 

With the same choice of axes, the components of the couple which is the 
equivalent of the fluid pressures on the solid, in the case of any uniform 
translation (Uy v, w), are 

L, M, N = (B - O) vw, (O - A) vm, (A - B) w (6) 

Hence if in the ellipsoid 

kx^ + By« + Cz^ = const., (7) 

we draw a radius vector r in the direction of the velocity (w, v, w) and erect 
the perpendicular h from the centre on the tangent plane at the extremity 
of r, the plane of the couple is that of h and r , its magnitude is proportional 
to sin {h, r)/h, and its tendency is to turn the solid in the direction from h to 
r. Thus if the direction of (i/, v, w) differs but slightly from that of the axis 
of X, the tendency of the couple is to diminish the deviation when A is the 
greatest, and to increase it when A is the least, of the three quantities A, B, C, 
whilst if A is intermediate to B and C the tendency depends on the position 
of r relative to the circular sections of the above elUpsoid. It appears then 
that of the three permanent translations one only is thoroughly stable, viz. 
that corresponding to the greatest of the three coefficients A, B, C. For 
example, the only stable direction of translation of an elUpsoid is that of its 
least axis; see Art. 121*. 

125. The above, although the simplest, are not the only steady motions 
of which the body is capable, under the action of no extraneous forces. The 
instantaneous motion of the body at any instant consists, by a well-known 
theorem of Kinematics, of a twist about a certain screw ; and the condition 
that this motion should be permanent is that it should not affect the 
configuration of the impulse (which is fixed in space) relatively to the body. 
This requires that the axes of the screw and of the corresponding impulsive 
wrench should coincide. Since the general equations of a straight line 
involve four independent constants, this gives four linear relations to be 
satisfied by the five ratios u : v : w : p : q : r. There exists then for every 
body, under the circumstances here considered, a singly-infinite system of 
possible steady motions. 

The steady motionfi next in importance to the three permanent translations are those 
in which the impulse reduces to a couple. The equations (1) of Art. 120 shew that we 
may have $, 7, f =0, and X, /x, v constant, provided 

\/p =fi/q = v/ry =1% say (1) 

If the axes of co-ordinates have the special directions referred to in the preceding Art., the 
conditions f, 17, f =0 give us at once u, v, to in terms of p, q, r, viz. 

Fp+rq + F'r, Qp+Q'q + G'r Hp+H'q+H''r ,., 

u^-^ ^ , t;= — ^ ^ , w= ^ ^ (2) 

* The physical cause of this tendency of a flat-shaped body to set itself broadside-on to the 
relative motion is clearly indicated in the diagram on p. 81. A number of interesting practical 
illustrations arc given by Thomson and Tait, Art. 325. 



124--125] Steady Motions 161 

Substituting these values in the expressions for X, thv obtained from Art. 122 (3), we find 

ae de de 
^^^""^Tp' 8^' ¥ <^) 

provided 2e (p, q, r) = 9p^-\-^ +Kr« +2Wgr +24®'rp +2Vi'pq, (4) 

the coefficients in this expression being determined by formulae of the types 

^ p F* G* H* ^, j^ F'F' Q'Q" WW ... 

These formulae hold for any case in which the force-constituent of the impulse is zero. 
Introducing the conditions (1) of steady motion, the ratios piqir are to be determined 
from the three equations 

9p +Vi'q ■>r&T =lcp, 

K>+€g +Wr = kq, (6) 

&P+Wq+Vit = kr. 

The form of these shews that the line whose direction-ratios are piqir must be parallel 
to one of the principal axes of the ellipsoid 

e (ar, y, z) =const (7) 

There are therefore three permanent screw-motions such that the corresponding impulsive 
wrench in each case reduces to a couple only. The axes of these three screws are mutually 
at right angles, but do not in general intersect. 

It may now be shewn that in all cases where the impulse reduces to a couple only, the 
motion can be completely determined. It is convenient, retaining the same directions of 
the axes, to change the origin. Now the origin may be transferred to any point (a;, y, z) 
by writing 

u+ry-qz, v+pz-rx, to+qx-py, 

for u, V, w respectively. The coefficient of 2rr in the expression for the kinetic energy, Art. 
121 (8), becomes -Bx+CT, that of 2wq becomes Cx +H\ and so on. Hence if we take 

^=lU-c> ^-^c-a)' '=Ka'b) W 

the coefficients in the transformed expression for 2T will satisfy the relations 

B~ C ' C A' A^B ^^^ 

If we denote the values of these pairs of equal quantities by a, /9, y respectively, the 
formulae (2) may be written 

IMt — — ^ "X , V -— "~ *s f V/ — ^ jj^ , .......a.....'. ..^ ^^} 

where 2* (p, q, r) =2 J>* + ;b ?* '^'C^ +2aqr+2prp +2y pq (11) 

The motion of the body at any instant may be conceived as made up of two parts ; viz. a 
motion of translation equal to that of the origin, and one of rotation about an instantaneous 
axis paiMing through the origin. Since (, 7, ^=0 the latter part is to be determined by 
the equations 

^=rM-gv, -^^P'-r^ 5=?X-PM. 

L. H. II 



162 Motion of Solids through a Liquid [chap, vi 

which express that the vector (X» /a, y) is constant in magnitude and has a fixed direction 
in space. Substituting from (3), 



(12) 



d 9e_ ae_ ae 

di 9g "^ ¥ '" 3p' 

d a9_ 9g_ ae 
(ft ar^^ap ^ag' 

These are identical in form with the equations of motion of a rigid body about a fixed 
point, so that we may make use of Poinsot's well-known solution of the latter problem. 
The angular motion of the body is obtained by making the ellipsoid (7), which is fixed in 
the body, roll on a plane 

Xa; + /jty + v2 = coast. , 

which is fixed in space, with an angular velocity proportional to the length 01 of the 
radius vector drawn from the origin to the point of contact /. The representation of the 
actual motion is then completed by impressing on the whole system of rolling ellipsoid 
and plane a velocity of translation whose components are given by (10). This velocity is 
in the direction of the normal OM to the tangent plane of the quadric 

* (a:, y, 2) = -«», (13) 

at the point P where 01 meets it, and is equal to 



0P,0M 



angular velocity of body (14) 



When 01 does not meet the quadric (13), but the conjugate quadric obtained by changing 
the sign of c, the sense of the velocity (14) is reversed*. 

126. The problem of the integration of the equations of motion of a solid 
in the general case has engaged the attention of several mathematicians, but, 
as might be anticipated from the complexity of the question, the physical 
meaning of the results is not easily grasped 1. 

In what follows we shall in the first place inquire what simplifications 
occur in the formula for the kinetic energy, for special classes of soUds, and 
then proceed to investigate one or two particular problems of considerable 
interest which can be treated without difficult mathematics. 

The general expression for the kinetic energy contains, as we have seen, 
twenty-one coefficients, but by the choice of special directions for the 
co-ordinate axes, and a special origin, these can be reduced to fifteen J. 

* The substance of this Art. is taken from a paper, " On the Free Motion of a Solid through 
an Infinite Mass of Liquid," Proc. Lond. Math. 8oc. t. viii. (1877). Similar results were 
obtained independently by Craig, "The Motion of a Solid in a Fluid,*' Amer. Joum. of Math. 
t. ii. (1879). 

f For references see Wien, Lehrbuch d. Hydrodynamik, Leipzig, 1900, p. 164. 

X Cf. Clebsoh, "Ueber die Bewegung eines Korpers in einer Fltissigkeit," Math. Ann. t. iii. 
p. 238 ( 1 870). This paper deals with the * reciprocal ' form of the dynamical equations, obtained by 
substituting from Art 122 (4) in Art. 120 (1). 



125-126] Hydrokinetic Symmetries 163 

The most symmetrical way of writing the general expression is 
2T = Au^ + Bv^ + Gw^ + 2A'vw ^2B'wu^ 20 'uv 
+ Pp^ + Qq^ + Rr^ + 2P'qr + 2Q'rp + 2R'pq 
+ 2Lup + 2Mvq + 22Vw 
+ 2-F (w + w^j) + 2G {wp + wr) + 2H (uq + vp) 
+ 2F' {vr - wq) + 2G' {wp - wr) + 2H' (uq - vp) (1) 

It has been seen that we may choose the directions of the axes so that 
A\ B\ C = 0, and it may easily be verified that by displacing the origin we 
can further make F\ G\ J?' = 0. We shall henceforward suppose these 
simpUfications to have been made. 

1°. If the sohd has a plane of symmetry, it is evident from the con- 
figuration of the relative stream-lines that a translation normal to this plane 
must be one of the permanent translations of Art. 124. If we take this 
plane as that of xy, it is further evident that the energy of the motion must 
be unaltered if we reverse the signs of w, p, q. This requires that P\ Q\ 
L, M, N^ H should vanish. The three screws of Art. 125 are now pure 
rotations, but their axes do not in general intersect. 

2"^. If the body has a second plane of symmetry, at right angles to the 
former one, we may take this as the plane xz. We find that in this case 
R' and G must also vanish, so that 

2T = Au^ + Bv^ + Cw^ + Pp^ + Qq^ + Rr^ -\-2F(vr + wq). . .(2) 

The axis of x is the axis of one of the permanent rotations, and those of the 
other two intersect it at right angles, though not necessarily in the same point. 

3°. If the body has a third plane of symmetry, say that of yz, at right 
angles to the two former ones, we have 

2T = Au^ + Bv^ + Cw^ + Pp^ -{-Qq^ + Rr^ (3) 

4°. Returning to (2°), we note that in the case of a solid of revolution 
about Oxy the expression for 2T must be unaltered when we write v, j, — w, — r 
for w, r, V, q, respectively, since this is equivalent to rotating the axes of y, z 
through a right angle. Hence 5 = (7, = i2, J = 0; and therefore 

2T = 4w« + 5 (v« + w^) + Pp^^Q {q^ + r«) (4)* 

The same reduction obtains in some other cases, for example when the 
sohd is a right prism whose section is any regular polygon f. This is seen at 
once from the consideration that, the axis of x coinciding with the axis of the 
prism, it is impossible to assign any uniquely symmetrical directions to the 
axes of y and z. 

* For the solution of the equations of motion in this case see Greenhill, "The Motion of a 
Solid in Infinite Liquid under no Forces," Amer. J, of Math, t. xx. (1897). 

t See Larmor, "On Hydrokinetic Symmetry," Quart. Joum. Math. t. xx. (1885). 

11—2 



164 Motion of Solids throiigh a Liquid [chap, vi 

5^. If, in the last case, the form of the solid be similarly related to each 

of the co-ordinate planes (for example a sphere, or a cube), the expression (3) 

takes the form 

2r = 4 (w2 ^- v2 + w;2) + p (p2 + ya ^_ ^2) (5) 

This again may be extended, for a like reason, to other cases, for example 
any regular polyhedron. Such a body is practically for the present purpose 
'isotropic,' and its motion will be exactly that of a sphere under similar 
conditions. 

6°. We may next consider another class of cases. Let us suppose that 
the body has a sort of skew symmetry about a certain axis (say that of x), 
viz. that it is identical with itself turned through two right angles about this 
axis, but has not necessarily a plane of symmetry*. The expression for 2T 
must be unaltered when we change the signs of v, w, q, r, so that the 
coefficients Q\ R\ G, H must all vanish. We have then 

2T « Au^ + Bv^ + Cw^ + Pp« + Qq^ + fir« + 2P'qr 

+ 2Lup -h 2Mvq + 2Nw + 2J (vr + wq) (6) 

The axis of a; is one of the directions of permanent translation ; and is also 
the axis of one of the three screws of Art. 125, the pitch being — LJA, The 
axes of the two remaining screws intersect it at right angles, but not in 
general in the same point. 

7°. If, further, the body be identical with itself turned through one 
right angle about the above axis, the expression (6) must be unaltered when 
V, J, — w;, — r are written for w, r, v, j, respectively. This requires that 
B = C, Q = fi, P' = 0, If = iV, J = 0. Hencet 

2T ==Au^-\-B {v^ + w^) + Pp^-\-Q (g* + r^) + 2Lup + 2M (vq ■\-wr), . . (7) 

The form of this expression is unaltered when the axes of y, z are turned 
in their own plane through any angle. The body is therefore said to possess 
hehcoidal symmetry about the axis of x, 

8^. If the body possess the same properties of skew symmetry about an 
axis intersecting the former one at right angles, we must evidently have 

2T^A (tt« + t;« + !£;«) -f- P (p* + ?« + r«) + 2L (j9w + ?v + rw). . .(8) 

Any direction is now one of permanent translation, and any hne drawn 
through the origin is the axis of a screw of the kind considered in Art. 125, 
of pitch — LjA, The form of (8) is unaltered by any change in the directions 
of the axes of co-ordinates. The solid is therefore in this case said to be 
'helicoidally isotropic' 

* A two-bladed screw-propeller of a ship is an example of a body of this kind. 

t This result admits of the same kind of generalization as (4), e,g. it applies to a body 
shaped like a screw-propeller with thrtt symmetrically- disposed blades. The integration of the 
equations of motion is discussed by Greenhill, "The Motion of a Solid in Infinite Liquid," 
Amer. J, of Maih. t. zxviii. p. 71 (1906). 



126-127] Solid of Revolution 165 

127. For the case of a solid of revolution, or of any other form to which 
the formula 

2T^ 4w« + 5 (v* + w^) + Pp« + Q (?* + r2) (1) 

applies, the complete integration of the equations of motion was effected by 
Ejrchhoff * in terms of elliptic functions. 

The particular case where the solid moves without rotation about its axis, 
and with this axis always in one plane, admits of very simple treatment !» 
and the results are very interesting. 

If the fixed plane in question be that of xy we have p, q, w = 0, so that 
the equations of motion, Art. 124 (1), reduce to 

* * , (2) 

Let z, y be the co-ordinates of the moving origin relative to fixed axes in 
the plane (xy) in which the axis of the soUd moves, the axis of z coinciding 
with the line of the resultant impulse (J, say) of the motion ; and let 6 be the 
angle which the line Ox (fixed in the solid) makes with z. We have then 

Au = I cos 0, Bv = — I BtaOy r = 6, 

The first two of equations (2) merely express the fixity of the direction of the 
impulse in space ; the third gives 

g6f + ^^^^^/*sinflcose = (3) 

« 

We may suppose, without loss of generality, that A> B, If we write 
2fl = ^, (3) becomes , 

^+-^^^— 8m& = 0, (4) 

which is the equation of motion of the common pendulum. Hence the 
angular motion of the body is that of a ' quadrantal pendulum,' i,e, a body 
whose motion follows the same law in regard to a quadrant as the ordinary 
pendulum does in regard to a half-circumference. When has been 
determined from (3) and the initial conditions, z, y are to be found from 
the equations 



/ J 

z = tt cos ^ — v sin fl = -1- cos* ^ + « sin* 6, 

y = w sin d + t; cosfl « f -J— ^ j sin fl cos d = y S, 



(5) 



* he ante p. 151. 

t See Thomson and Tait, Art. 322; Qreenhill, "On the Motion of a Cylinder through a 
Friotionless Liquid under no Forces,*' Mea9, of Math, t. ix. (1880). 



166 Motion of Solids throtigh a Liquid [chap, vi 

the latter of which gives 

l-jO, (6) 

and is otherwise obvious, the additive constant being zero since the axis of z 
is taken to be coincident with, and not merely parallel to, the line of the 
impulse /. 

Let us first suppose that the body makes complete revolutions, in which 
case the first integral of (3) is of the form 

fi« = a>2 (1 - ** sin* e), (7) 

^^^^^^ *'==4w-S (^) 

Hence, reckoning t from the position fl = 0, we have 



= f ' — T = F(k,0),.: (9) 



cot 

in the usual notation of elUptic integrals. If we eUminate t between (5) and 
(7), and then integrate with respect to 0, we find 

the origin of z being taken to correspond to the position 6 = 0. The path 
can then be traced, in any particular case, by means of Legendre's Tables. 
See the curve marked I on the opposite page. 

If, on the other hand, the solid does not make a complete revolution, but 
oscillates through an angle a on each side of the position ^ = 0, the proper 
form of the first integral of (3) is 

d^ = a>>(l-'^l^) (11) 

where sin* a = -j — -^ • ■?« (12) 

A — B P 

If we put sin ^ = sin a sin 0, 

2 

this gives -^^ = ^-g-- (1 — sin* a sin*0), 

whence -= — = F (sin a, 0) (13) 

sma ^ 

Transforming to as independent variable, in (5), and integrating, we find 

IT ^\ 

z = -^- sin a . -P (sin a, 0) f- cosec a . E (sin a, 0), | 

^ ^ I. ...(14) 

y = -V cos ^p. 



127] 



Solid of Revolution 



167 



The path of the point is now a sinuous curve crossing the line of the 
impulse at intervals of time equal to a half-period of the angular motion. 
This is illustrated by the curves III and IV of the figure. 




There remains a critical case between the two preceding, where the solid 
just makes a half-revolution, having as asymptotic limits the two values 



4 

168 Motion of Solids throtigh a Liquid [chap, vi 

± Jtt. This case may be obtained by putting A; = 1 in (7), or a = Jtt in (11) ; 
and we find 

d' = ai 008 d, ; (15) 

ad = log tan (Jtt + ^6), (16) 

= D- log ^^ (i^ + i^) - ^ sin 0, 
^ ^ y (17) 



z = 



y = ^ cos 0, 

See the curve II of the figure*. 

It is to be observed that the above investigation is not restricted to 
the case of a soHd of revolution; it applies equally well to a body with 
two perpendicular planes of symmetry, moving parallel to one of these 
planes, provided the origin be properly chosen. If the plane in question be 
that of xy, then on transferring the origin to the point {F/B, 0, 0) the last 
term in the formula (2) of Art. 126 disappears, and the equations of motion 
take the form (2) above. On the other hand, if the motion be parallel to zx 
we must transfer the origin to the point (— F/C, 0, 0). 

The results of this Article, with the accompanying diagram, serve to 
exemplify the statements made near the end of Art. 124. Thus the curve IV 
illustrates, with exaggerated amplitude, the case of a sUghtly disturbed stable 
steady motion parallel to an axis of permanent translation. The case of 
a shghtly disturbed unstable steady motion would be represented by a curve 
contiguous to II, on one side or the other, according to the nature of the 
disturbance. 

128. The mere question of the stabiUty of the motion of a body parallel 
to an axis of symmetry may of course be more simply treated by approximate 
methods. Thus, in the case of a body with three planes of symmetry, as in 
Art. 126, 3°, sUghtly disturbed from a state of steady motion parallel to a;, we 
find, writing u = Uq-\- u\ and assuming u\ v, w, p, q, r to be all small. 






^ ....(1) 



* In order to bring out the peculiar features of the motion, the curves have been drawn for 
the somewhat extreme case of A =5B. In the case of an infinitely thin disk, without inertia of 
its own, we should have A/B=co ; the curves would then have cusps where they meet the 
axis of y. It appears from (5) that ± has always the same sign, so that loops cannot occur in 
any case. 

In the various cases figured the body is projected always with the same impulse, but with 
different degrees of rotation. In the curve I, the maximum angular velocity is ^2 times what 
it is in the critical case 11 ; whilst the curves III and IV represent oscillations of amplitude 45^ < 

and IS*' respectively. 



127-129] Solid of Revolution 169 

Hence ^ A» "^ ^W — "•* » = 0, 

with a Bimilar equation for r, and 

^i*w A{A-C) ^ - ,„, 

C-^ + — ^-g ^V«'=0. (2) 

with a similar equation for q. The motion is therefore stable only when A 
is the greatest of the three quantities A, B, C. 

It is evident from ordinary Dynamics that the stability of a body moving parallel to an 
axis of symmetry will be increased, or its instability (as the case may be) wUl be diminished, 
by communicating to it a rotation about this axis. This question has been examined by 
Greenhill*. 

Thus, in the case of a solid of revolution slightly disturbed from a state of motion in 
which u and p are constant and the remaining velocities are zero, if we neglect squares 
and products of small quantities the first and fourth of equations (1) of Art. 124 give 

du/cU=0, dp/dt=0, 

whence «=«0' P=Po* (3) 

say, where Uq, p^ are constants. The remaining equations then take, on substitution from 
Art. 126 (3), the forms 

J^r^-l>o«^j=-^V» B\^ + p^v\=Au^ (4) 

Q^HP'Q)P^=-(A-B)u^w, Q^-{P-Q)p^^{A-B)UoV. ....(6) 

If we assume that v, tr, g, r vary as e*'*, and eliminate their ratios, we find 

e<r«±(P-2G)l>o<r-{(P-G)V+^(^-5)V}=0. (6) 

The condition that the roots of this should be real lb that 

should be positive. This is always satisfied when A > B, and can be satisfied in any case 
by giving a sufficiently great value to p^. 

This example illustrates the steculiness of fiight which is given to an elongated projectile 
by rifling. 

129. In the investigation of Art. 125 the term 'steady' was used to 
characterize modes of motion in which the 'instantaneous screw' preserved 
a constant relation to the moving sohd. In the case of a solid of revolution, 
however, we may conveniently use the term' in a somewhat wider sense, 
extending it to motions in which the vectors representing the velocities 
of translation and rotation are of constant magnitude, and make constant 
angles with the axis of symmetry and with each other, although their relation 
to points of the solid not on the axis may continually vary. 

* "Fiiiid Motion between Confocal Elliptic Cylinders, ^c." Quart. Joum. Math. t. zvi. (1879). 



170 Motion of Solids throtigh a Liquid [chap, vi 

The conditions to be satisfied in this case are most easily obtained from the equations 
of motion of Art. 124, which become, on substitution from Art. 126 (4), 

A%=B(rv.^), Pf=0. \ 
B^^Bjnv-Am, Q^^^(A-B)uw- P-Q)pr,\ (1) 

B^^^Agu-Bpv, gJ= (A-B)uv + (P-Q)n'^ 

It appears that j} is in any case constant, and that g' +r* will also be constant provided 

v/q=tD/r, =*, say (2) 

This makes dufdt =0, and v" +t&' = const. It follows that k will also be constant; and it 
only remains to satisfy the equations 

kB^=(kBp-Au)r, Q^= -{{A -B)ku+{P-Q)p}r, 

These will be consistent provided 

kB {{A -B) ku + {P -Q) p) +Q {kBp -Au)=0, 

, u kBP .^. 

^^'^'^^^ p=AQ-k^B(A-B) <^) 

Hence by variation of k we obtain an infinite number of possible modes of steady motion, 
of the kind above defined. In each of these the instantaneous axis of rotation and the 
direction of translation of the origin are in one plane with the axis of the solid. It is 
easily seen that the origin describes a helix about the line of the impulse. 

These results are due to Kirchhoff. 

130. The only case of a body possessing helicoidal property, where 
simple results can be obtained, is that of the 'isotropic helicoid' defined by 
Art. 126 (8). Let be the centre of the body, and let us take as axes of 
co-ordinates at any instant a line Ox parallel to the axis of the impulse, 
a line Oy drawn outwards from this axis, and a line Oz perpendicular to the 
plane of the two former. If I and K denote the force- and couple-constituents 
of the impulse, we have 

Au + Lp= f = /, Av-i- Lq = 7) =0y Aw + Lr = ^ = 0, 
Pp -^ Lu = X = K, Pq + Lv = fjL = 0, Pr -{- Lw = v = Iw, 

where w denotes the distance -of from the axis of the impulse. 

Since AP — L^^Oy the second and fifth of these equations shew that v = 0, 
q = 0. Hence m is constant throughout the motion, and the remaining 
quantities are also constant; in particular 

PI - LK LIw 



= 1/ = JrnJ 



129-132] Helicoidal Symmetry 171 

The origin therefore describes a helix about the axis of the impulse, 
of pitch 

I L' 

This example is due to Kelvin*. 

a 

131. Before leaving this part of the subject we remark that the 
preceding theory applies, with obvious modifications, to the acyclic motion 
of a liquid occupying a cavity in a moving solid. If the origin be taken at 
the centre of inertia of the liquid, the formula for the kinetic energy of the 
fluid motion is of the type 

2T = m (w2 + v« + w^) + Pp* + Q?* + Rr« + 2V'qr + 2QVp + ^Kjq. . . (1) 

For the kinetic energy is equal to that of the whole fluid mass (m), supposed 
concentrated at its centre of inertia and moving with this point, together with 
the kinetic energy of the motion relative to the centre of inertia. The latter 
part of the energy is easily proved by the method of Arts. 118, 121 to be 
a homogeneous quadratic function of j9, q, r . 

Hence the fluid may be replaced by a solid of the same mass, having the 
same centre of inertia, provided the principal axes and moments of inertia be 
properly assigned. 

The values of the coefficients in (1), for the case of an ellipsoidal cavity, may be calcu- 
lated from Art. 110. Thus, if the axes of x, y, z coincide with the principal axes of the 
ellipsoid, we find 

p. ,.».,» '-55$1, ,»<$=^'. ^^^■. p-.*,E'=a 

Case of a Perforated Solid, 

132. If the moving solid have one or more apertures or perforations, so 
that the space external to it is multiply-connected, the fluid may have 
a motion independent of that of the solid, viz. a cyclic motion in which the 
circulations in the several irreducible circuits which can be drawn through 
the apertures may have any given constant values. We will briefly indicate 
how the foregoing methods may be adapted to this case. 

* Lc. ante p. 158. It is there pointed out that a solid of the kind here in question may be 
constructed by attaching vanes to a sphere, at the middle points of twelve quadrantal arcs drawn 
BO as to divide the surface into octants. The vanes are to bo perpendicular to the surface, and 
are to be inclined at angles of 45° to the respective arcs. Larmor {l.c. ante p. 163) gives another 
example. **If . . .we take a regular tetrahedron (or other regular solid), and replace the edges 
by skew bevel faces placed in such wise that when looked at from any comer they all slope the 
same way, we have an example of an isotropic helicoid.** 

For some further investigations in the present connection sec a paper by Biiss Fawcett, "On 
the Motion of Solids in a Liquid," Quart. Jonm. Math. t. xxvi. (1893). 



172 Motion of Solids through a Liquid [chap, vi 

Let #c, #c', #c", ... be the circulations in the various circuits, and let So*, So*', 
Sot", ... be elements of the corresponding barriers, drawn as in Art. 48. 
Further, let I, m, w denote the direction-cosines of the normal, drawn towards 
the fluid at any point of the surface of the sohd, or drawn on the positive 
side at any point of a barrier. The velocity-potential is then of the form 

where ^ = ^i + t;^« + «^s + ;%i + 8X2 + rXs.l qj 

The functions ^i> ^2> ^S) Xi> Xs> Xs ^^^ determined by the same conditions as 
in Art. 118. To determine co, we have the conditions : (1°) that it must 
satisfy V\o = at all points of the fluid ; (2°) that its derivatives must vanish 
at infinity ; (3°) that dw/dn must = at the surface of the solid ; and (4^) that 
CO must be a cyclic function, diminishing by unity whenever the point to which 
it refers completes a circuit cutting the first barrier once (only) in the positive 
direction, and recovering its original value whenever the point completes a 
circuit not cutting this barrier. It appears from Art. 52 that these conditions 
determine co save as to an additive constant. In like manner the remaining 
functions co', co", ... are determined. 

By the formula (5) of Art. 55, twice the kinetic energy of the fluid is 
equal to 

-pjj{<f>+<f>o)^{<f> + <f>o)d8 

- pf^jj g^ (^ + ^0) *y - P*^' j^i^ +<f>o)da'- (2) 

Since the cycUc constants of <f> are zero, and since d<f>QJdn vanishes at the 
surface of the solid, we have, by Art. 54 (4), 



M^+'//l*'+"'//l!^+ •••-//* 



ysds-o. 

on 



Hence (2) reduces to 

-,jl^^iS-^jj%'i,-^'ij^i,'- (3) 

Substituting the values of ^, ^0 from (1) we find that the kinetic energy 
of the fluid is equal to 

T + Z, (4) 

where T is a homogeneous quadratic function of w, v, w, p, y, r, of the form 
defined by Art. 121 (2) (3), and 

2K = (#c, k) k^ + (#c', #c') k'« + . . . + 2 {k, k')kk' + ..., . . .(6) 



V (6) 



13^133] Perforated Solid 173 

where, for example, 

{k, k)=-p fj^ da, 

The identity of the different forms of {k, k) follows from Art. 64 (4). 
Hence the total energy of fluid and solid is given by 

T = ® + ii:, (7) 

where 'ST is a homogeneous quadratic function of Uy v, w, p, q, r of the same 
form as Art. 121 (8), and K is defined by (5) and (6) above. 

133. The 'impulse' of the motion now consists partly of impulsive forces 
applied to the sohd, and partly of impulsive pressures pK, pK, pK*\ . . . applied 
uniformly (as explained in Art. 54) over the several membranes which are 
supposed for a moment to occupy the positions of the barriers. Let us 
denote by ^i, 171, Jj, Aj, ftj, i/j the components of the extraneous Impulse 
applied to the solid. Expressing that the x-component of the momentum of 
the solid is equal to the similar component of the total impulse acting on it, 
we have 

= ii + P M(«^i + • • • + PXi+ "• + f<" + ...) S- dS 

where, as before, Tj denotes the kinetic energy of the solid, and T that part 
of the energy of the fluid which is independent of the cyclic motion. Again, 
considering the angular momentum of the solid about the axis of x, 

1 = Ai - pjj((f> + ^0) (wy - mz) dS 






(5) 



174 Motion of Solids through a Liquid [chap, vi 

Hence, since "ST = T + T^, we have 

*.-f-'«//"i'^-w/"'i^--- 

By virtue of Lord Kelvin's extension of Green's Theorem, already referred 
to, these may be written in the alternative forms 

Adding to these the terms due to the impulsive pressures applied to the 
barriers, we have, finally, for the components of the total impulse of the 
motion*, 

where, for example, 

^•-^■//('+l')*'+^'//('+^')*''+-. 

It is evident that the constants lo> 'yo* ^o> ^, /^o> ^o are the components 
of the impulse of the cychc fluid motion which would remain if the solid 
were, by forces applied to it alone, brought to rest. 

By the argument of Art. 119, the total impulse is subject to the same 
laws as the momentum of a finite d3mamical system. Hence the equations 
of motion of the solid are obtained by substituting from (5) in the equations 
(1) of Art. 120t. 

134. As a simple example we may take the case of an annular solid of 
revolution. If the axis of x coincide with that of the ring, we see by 
reasoning of the same kind as in Art. 126, 4° that if the situation of the 
origin on this axis be properly chosen we may write 

2T = Au^-hB {v^ + w^) + Pp^-hQ {q^ + r«) + (k, k) kK . . .(1) 

Hence f , 77, ^ = .iw + fo, B% ^^l A, ^, 1/ = Pp, Qq,Qr (2) 

* Cf. Sir W. Thomson, Ic. ante p. 158. 

t This conclusion may be verified by direct calculation from the preesure-formula of Art. 20; 
see Bryan, " Hydrodynamical Proof of the Equations of Motion of a Perforated Solid, 
Phil Mag. May 1893. 



da' -h 



• ..(6) 



»» 



J 



133-135] Components of Impvlse lib 

Substituting in the equations of Art. 120, we find dpldt = 0, or j? = const., 
as is otherwise obvious. Let us suppose that the ring is slightly disturbed 
from a state of motion in which v, w, p, q, r are zero, i.e. a steady 
motion parallel to the axis. In the beginning of the disturbed motion 
V, Wy p, q, r will be small quantities whose products we may neglect. The 
first of the equations referred to then gives du/dt = 0, or w = const., and the 
remaining equations become 

B^ = -(Au + Ur, Q^=-{iA-B)u + io}^^ 

B^= {Au + io)q, g|= {{A-B)u + Qv.\ 

Eliminating r, we find 

BQ^, = -iAu + ^o){{A-B)u + i^}v. ., (4) 

Exactly the same equation is satisfied by w. It is therefore necessary and 
sufficient for stability that the coefficient of v on the right-hand side of (4) 
should be negative ; and the time of a small oscillation, when this condition 
is satisfied, is* 



g^ r SQ -li 



(5) 



We may also notice another cane of steady motion of the ring, viz. where the impulse 
reduces to a couple about a diameter. It is easily seen that the equations of motion are 
satisfied by (, i;, (, X, /a =0, and v constant ; in which case 

u= -(o/A^ r= const. 

The ring then rot-ates about an axis in the plane yz parallel to that of z, at a distance u/r 
from itf. 

Equation of Motion in Generalized Co-ordinates, 

135. When we have more than one moving soUd, or when the fluid is 
bounded, wholly or in part, by fixed walls, we may have recourse to Lagrange's 
method of * generahzed co-ordinates.' This was first apphed to hydrodynamical 
problems by Thomson and TaitJ. 

In any dynamical system whatever, if ^, rj, J are the Cartesian co-ordinates 
at time t of any particle m, and X, Y, Z the components of the total force 
acting on it, we have of course 

mf=Z, mrj=Y, ml=^Z (1) 

♦ Sir W. Thomson, l.c, ante p. 168. 

t For further investigations on this subject we refer to papers by Basset, *'0n the Motion 
of a Ring in an Infinite Liquid," Proc. Camb, Phil. Soc. t. vi. (1887), and Miss Fawcott, Ix. ante 
p. 171. 

; Natural Philowpky (Ist ed.), Oxford, 18A7, Art. 331. 



176 Motion of Solids throv^h a ^Liquid [chap, vi 

Now let ^ + A^, ^i + Aiy, J + AJ be the co-ordinates of the same particle, at 
time f, in any arbitrary motion of the system differing infinitely little from 
the actual motion, and let us form the equation 

Sw (^Af + iyAiy + t'AO = S (ZA^ + YLri + ZAO, (2) 

where the summation I! embraces all the particles of the system. This 
follows at once from the equations (1), and includes these, on account of the 
arbitrary character of the variations Af , At;, A J. Its chief advantages, how- 
ever, consist in the extensive elimination of internal forces which, by imposing 
suitable restrictions on the values of Af , At;, A^, we are able to effect, and in 
the facilities which it affords for transformation of co-ordinates. It is to be 
noticed that 

SO that the symbols d and A are commutative. 

The systems ordinarily contemplated in Analytical Dynamics are of 
finite freedom; i.e. the position of every particle is completely determined 
when we know the values of a finite number of independent variables or 
'generalized co-ordiiiates' j'l, j^a* • • • in^^^ that, for example, 

. 3^ . _^ a^ . _^ _^ a^ . 
^^ai;^^"'a^'^-'----'arn*- ^ ^^ 

The kinetic energy can then be expressed as a homogeneous quadratic 
function of the 'generaUzed velocity-components' ^i, ft* • • • ?n> thus 

2r = ii^gi' + ^229t* + . . . + 2iii,?ig2 + . . . , (4) 

where 

(5) 

The quantities ii^^, -4„ are called the 'inertia-coefficients' of the system; 
they are in general functions of the co-ordinates y^, y,* ••• ?n- 

Again, we have 

2 (X^^ + Y^r| + ZAO = g^A?, + Q^^q^ + . . . + Qn^qn. • • • .(6) 

-k- 0,.2(x|+r| + z|) ,7) 

The quantities Q^ are called the 'generalized components of force.' In the 
case of a conservative system we have 

«— 1^, <») 



135] Generalized Co-ordinates 177 

Also, from (3) and (5), 
2m (^A^ + ^At; + ^AO = (^iiffi + ^i%q% + . . . + A^nin) A?i 

4- 

+ Mnl?! + -4n2?2 + • • • + A^^q^) Ajn 

= a^,^«^ + a^/**-^---+3f/?- ••••(^) 

or Sm (^Af + aJAt; + ^A^ = ^i A?i + pj Aj, + . . . + ?„ A?„, (10) 

where P^^dd ^^^^ 

The quantities Pr are called the 'generalized components of momentum' of 
the system. When T is expressed as in (4) as a homogeneous quadratic 
function of g^, g^* • • • 9«> we have 

22' = Ptqi + Ptqt + . . . + Pnqn (12) 

Since 2m (^A^ 4- rj^rj + gAO = | 2m (f A^ + ^Ar? + ^AO - AT, . . (13) 

the transformation of (2) to generalized co-ordinates is easily completed by 
substitution from (9) and (6). The variations A^^ of the velocities cancel; 
and, equating coefficients of the independent variations ^q^ of the co-ordinates, 
we obtain n equations of the type* 

ddT_dT_ 

From (12) and (14) we derive 

2 ^ = ftjj + Pxqx + p^qt + p^q^ + . . . + /»«?« + Pnqn 

+ o~ ?! + 5 — ?« + • • • + o" W« 
= -^ + Cl?l + 02?2 + • . . + ©«3n, 

whence -^ = Q^q^ -t- 0a92 + . . . + Q„9n, ; (15) 

or, in the case of a conservative system 

|(r+7) = 0. (16) 

which is the equation of energy. 

* This somxnary of Lagrange's proof ia introduced merely to facilitate reference to the 
▼arions steps, in the hydrodynamical investigation of the next Art. A proof by direct transforma- 
tion of co-ordinates, not involving the method of 'variations/ has been given by Hamilton {PhxL 
Trans. 1835, p. 96), Jacobi, Bertrand, and Thomson and Tait; see also Whittaker, Anal^ical 
Dynamics, Cambridge, 1904, p. 33. 

L.H. 12 



178 Motion of Solids throttgh a Liquid [chap, vi 



If we multiply (2) by &, and integrate between the limits t^ and t^ , we 
find, having regard to (13), 

''{AT + 2 {X£ii + YArj -f ZAO} dt = 



/ 



-(17) 

If we now introduce the additional condition that in the varied motion 
the initial and final positions shall be respectively the same for each particle 
as in the actual motion, the quantities A^, A77, A^ will vanish at both limits, 
and the equation reduces to 

f ' {AT -f 2 (ZA^ + TAt? + ZAO} * = 0, (18) 



or, for a conservative system*, 






(19) 



In words, if the actual motion of the system between any two configura- 
tions through which it passes be compared with any slightly varied motion, 
between the same configurations, which the system is (by the application of 
suitable forces) made to execute in the same time, the time-integral of the 
'kinetic potential' | V — T is stationary. 

In terms of generalized co-ordinates, the equation (18) takes the form 

' (AT + GiAji + GaAy, + . . . + Qn^qn) dt = (20) 

This embraces the whole dynamics of the system in a mathematically 
compact form. From it Lagrange's equations can immediately be deduced ; 
cf. Art. 139. 



/: 



136. Proceeding now to the hydrodynamical problem, let ji, jj, . . . j„ 
be a system of generalized co-ordinates which serve to specify the configuration 
of the solids. We will suppose, for the present, that the motion of the fluid 
is entirely due to that of the soUds, and is therefore irrotational and acyclic. 

In this case the velocity-potential at any instant will be of the form 

<f> = qi<f>i + q2<f>2 + . . . + qn<f>n9 (1) 

where <f>i,<f>%^ ... are determined in a manner analogous to that of Art. 118. 
The formula for the kinetic energy of the fluid is then 



2T 



= - P \w g^ ^'^ = -A-nji* -f A2292* -f . . . 4- 2A129392 + . . . , . (2) 



♦ Sir W. R. Hamaton, "On a General Method in Dynamics," PhiL Trans, 1834, 1836. 

t The name was introduced by Helmholtz, "Die physikalische Bedeutung des Pnncips der 
kleinsten Wirkung," CreUe, t. c. p. 137 (1886) [Wiss, Abh. t. iii. p. 203]. Whittaker, Analytical 
Dynamics, p. 38, reverses the sign. 



135-136] Application to Hydrodynamics 179 

where 

A„=-p|)V,^J^'<iS. A„^ - p jf<f>r^^dS=^ - p jj<f>,^-^dS, ..(3) 

the integrations extending over the instantaneous positions of the bounding 
surfaces of the fluid. The identity of the two forms of A„ follows from 
Green's Theorem. The coefficients A^^, A„ will in general be functions of 
the co-ordinates 9i, 9'2> • • - 9n* 

If we add to (2) twice the kinetic energy, Ti, of the solids themselves, we 
get an expression of the same form, with altered coefficients, say 

2T = ^nSi* -f ^22?2* + . . . + 2A^^q^q^ + (4) 

It remains to shew that, although our system is one of infinite freedom, 
the equations of motion of the solids can, under the circumstances pre- 
supposed, be obtained by substituting this value of T in the Lagrangian 
equations, Art, 135 (14). We are not at liberty to assume this without 
further examination, for the positions of the various particles of the fluid are 
not determined by the instantaneous values yi, ?a, . . . J« of the co-ordinates 
of the solids. For instance, if the solids, after performing various evolutions, 
return each to its original position, the individual particles of the fluid will 
in general be found to be finitely displaced*. 

Going back to the general formula (2) of Art. 135, let us suppose that in 
the varied motion, to which the symbol A refers, the solids undergo no 
change of size or shape, and that the fluid remains incompressible, and has, 
at the boundaries, the same displacement in the direction of the normal as 
the solids with which it is in contact. It is known that under these 
conditions the terms due to the internal reactions of the solids will disappear 
from the sum 

S (ZAf -f YArj + ZAO. 

The terms due to the mutual pressures of the fluid elements are equivalent to 

or |]y VM + "A, + nAJ) *S + Jj Jy (^ + ^ + ?^) i« JjA, 

where the former integral extends over the bounding surfaces, and I, m, n 
denote the direction-cosines of the normal, drawn towards the fluid. The 
volume-integral vanishes by the condition of incompressibility 

W'^'d^^ dz "^ (^^ 

* As a simple example, take the case of a circular disk which is made to move, without 
rotation, so that its centre describes a rectangle two of whose sides are normal to its plane; and 
examine the displacements of a particle initially in contact with the disk at its centre. 

12—2 



180 Motion of Solids through a Liquid [chap, vi 

The surface-integral vanishes at a fixed boundary, where 

and in the case of a moving solid it is cancelled by the terms due to the 
pressure exerted by the fluid on the solid. Hence the symbols Xj F, Z may 
be taken to refer only to the extraneous forces acting on the system, and we 
may write 

2 {X^^ -f FAt? + ZAO = Qi Aft + Q, Aft + . . . + QMn, .... (6) 

where Qi, ^2* . • . Cn denote the generalized components of extraneous force. 

The varied motion of the fluid has still a high degree of generality. We 
will now further limit it by supposing that while the soUds are, by suitable 
forces applied to them, made to execute an arbitrary motion, the fluid is left 
to take its own course in consequence of this. The varied motion of the 
fluid may accordingly be taken to be irrotational, in which case the varied 
kinetic energy T + AT of the system will be the same function of the 
varied co-ordinates q^ -f Aj,., and the varied velocities q^ + Aj^ , that the actual 
energy T is of j^ and g,.. 

Again, considering the particles of the fluid alone, we shall have, on the 
same supposition, 

= p jj<f> (IA| + mArj + nAO dS, 

where use has again been made of the condition (5) of incompressibility. By 
the kinematical condition to be satisfied at the boundaries, we have 

lAi + mArf -f nAf = - -^^ Aft - ^ Aft - . . . - -J^ Aj„, 
and therefore 

2m(|Af.fi7A^4-eA0 = ^p||<^(^^^Aft-f^*Aft-f...-h^^ 

= (Aiift + Aijft 4- . . . -f Aj„4„) Aft -f (Asift + Ajjft + . • + A^n^n) Aft 

-f . . . + (A„ift -f A„2ft + . . . + A„ngn) Ay„ 

9T 3T 3T 

= a^,^?^ + g^,^?* + '--+a9/?"' (^) 

by (1), (2), (3) above. If we add the terms due to the solids, we find that 
the relation (9) of Art. 135 still holds; and the deduction of Lagrange's 
equations 

dtdi.'d^,-^' ^^' 

then proceeds exactly as before. 



136-137] Application to Hydrodynamics 181 

As in Art. 135, these equations lead to 

or, in the case of a conservative system, 

r + F = const. (9) 

137. As a first application of the foregoing theory we may take an 
example given by Thomson and Tait*, where a sphere is supposed to move 
in a Uquid which is limited only by an infinite plane wall. 

Taking, for simpUcity, the case where the centre moves in a plane 
perpendicular to that of the wall, let us specify its position at time t by 
rectangular co-ordinates x, y in this plane, of which y denotes distance from 
the wall. We have 

2T = Ax*^By\ (1) 

where A and B are functions of y only, it being plain that the term iy 
cannot occur, since the energy must remain unaltered when the sign of i; is 
reversed. The values of A, B can be written down from the results of 
Arts. 98, 99, viz. if m denote the mass of the sphere, and a its radius, 
we have 



A = m-\- 



l-npa* (l + A^). -B = »» + I'rpo* (l + I ^). . .(2) 

approximately, if y be great in comparison with a. 
The equations of motion give 

im-x. >^,-K|^,|^).y. ,3, 

where X, Y are the components of extraneous force, supposed to act on the 
sphere in a line through the centre. 

If there be no extraneous force, and if the sphere be projected in a 
direction normal to the wall, we have ^ = 0, and 

By^ = const (4) 

Since B diminishes as y increases, the sphere experiences an acceleration 
from the wall. 

Again, if the sphere be constrained to move in a line parallel to the wall, 
we have y = 0, and the necessary constraining force is 

^=-if^ (5) 

* Ix, ante p. 175. 



I 
t 

i 

I 
t 



(2) 



182 Motion of Solids through a Liquid [chap, vi 

Since dAjdy is negative, the sphere appears to be attracted by the wall. The 
reason of this is easily seen by reducing the problem to one of steady motion. 
The fluid velocity will evidently be greater, and the pressure therefore less, 
on the side of the sphere next the wall than on the further side; see 
Art. 23. 

The above investigation will also apply to the case of two spheres 
projected in an unlimited mass of fluid, in such a way that the plane y = 
is a plane of symmetry in all respects. 

138. Let us next take the case of two spheres moving in the line 
of centres. 

The kinematical part of this problem has been treated in Art. 98. If we now denote 
by X, y the distances of the centres of the spheres A^ B from some fixed origin in the line 
joining them, we have 

2T=Lx^-2Mxy +Ny^ (1) 

where the coefficients L, M, N are functions of y -x, or c, the distance between the 
centres. Hence the equations of motion are 

where X, Y are the forces acting on the spheres along the line of centres. If the radii a, b 
are both smaU compared with c, we have, by Art. 98 (16), keeping only the most important 
terms, 

Z=m+f7rpa", Jbr=2»rp— 3-, N=m'+^npb^, (3) 

approximately, where m, m' are the masses of the two spheres. Hence to this order of 
approximation 

dL ^ dM ^ a»6» dN ^ 

Tc=^^ -d^=-^''P-^' d^=^- 

If each sphere be constrained to move with constant velocity, the force which must be 
applied to ^ to maintain its motion is 

^ dM dM .. _ a*6' .• ,.. 

^=-a^yiy-^)-S^'»9=«^P-^i/'- (4) 

This tends towards B, and depends only on the velocity of B. The spheres therefore 
appear to repel one another ; and it is to be noticed that the apparent forces are not equal 
and opposite unless x= ±,y. 

Again, if each sphere make small periodic oscillations about a mean position, the period 
being the same for each, the mean values of the first terms in (2) will be zero, and the 
spheres therefore will appear to act on one another with forces equal to 

6«-p -^ Wl (5) 

where \xy] denotes the mean value of xy. Ji &, y differ in phase by less than a quarter- 
period, this force is one of repulsion, if by more than a quarter-period it is one of attraction. 



137-139] Cydic Motion 183 

Next, let B perform small periodic oscillations, while A is held at rest. The mean force 
which must be applied to jii to prevent it from moving is 

J=4'^[y*] («) 

where [y*] denotes the mean square of the velocity of B, To the above order of approxi- 
mation dN/dc \b zero ; on reference to Art. 98 we find that the most important term in it 
is - I2ir pa^b^c'f so that the force exerted on il is attractive, and equal to 

«Tp-,f OT (7) 

This result comes under a general principle enunciated by Kelvin. If we have two 
bodies immersed in a fluid, one of which (A) performs small vibrations while the other {B) 
is held at rest, the fluid velocity at the surface of B will on the whole be greater on the 
side nearer A than on that which is more remote. Hence the average pressure on the 
former side will be less than that on the latter, so that B will experience on the whole an 
attraction towards A. As practical illustrations of this principle we may cite the apparent 
attraction of a delicately-suspended card by a vibrating tuning-fork, and other similar 
phenomena studied experimentaUy by Guthrie* and explained in the above manner by 
Kelvin f. 



Modification of Lagrange^s Equalions in the case of Cyclic Motion, 

139. We return to the investigations of Art. 135, with the view of 
adapting them to the case where the fluid has cyclic irrotational motion 
through channels in the moving solids, or (it may be) in an enclosing 
vessel, independently of the motion due to the soUds themselves. 

Let us imagine barrier-surfaces to be drawn across the several apertures. 
In the case of channels in a containing vessel we shall suppose these ideal 
surfaces to be fixed in space, and in the case of channels in a moving solid 
we shall suppose them to be fixed relatively to the soUd. Let 'j(, ;^', x'\ . . . 
be the fluxes at time t across, and relative to, the several barriers; and let 
Xi x'» x"? • • • be the time-integrals of these fluxes, reckoned from some 
arbitrary epoch, these quantities determining (therefore) the volumes of 
fluid which have up to the time t crossed the respective barriers. It will 
appear that the analogy with a dynamical system of finite freedom is still 
conserved, provided the quantities x, x'. x". .-be reckoned as generaUzed 
co-ordinates of the system, in addition to those (ji, ^2* • • • ?n) which specify 
the positions of the moving solids. It is obvious already that the absolute 
values of Xj x'» x"> • • • ^^^ ^^^ enter into the expression for the kinetic 
energy, but only their rates of variation. 

♦ "On Approach caused by Vibration," Proc, Boy. 8oc, t. xix. (1860) [PA*/. Mag. Nov. 1870]. 

t Beprint of Papers on Electrogtatics, Ac. Art. 741. For references to further investigations, 
both experimental and theoretioal, by Bjerknes and others, on the mutual infiaence of oscillating 
spheres in a fluid, see Hicks, "Report on Recent Researches in Hydrodynamics," Brii. Am. Rep. 
1882, pp. 52. . . ; Love, BncycL d, math, Wies. U iv. (3). pp. Ill, 112. 



184 Motion of Solids through a Liquid [chap, vt 

In the first place, we may shew that the motion of the fluid, in any given 
configuration of the solids, is completely determined by the instantaneous 

values of ^1, jjj • • • 9n> X» X^ ic'^ ^^^ ^' there were two modes of 

irrotational motion consistent with these values, then, in the motion which 
is the difference of these, the boundaries of the fluid would be at rest, and 
the flux across each barrier would be zero. The formula (5) of Art. 55 shews 
that under these conditions the kinetic energy must vanish. 

It follows that the velocity-potential can be expressed in the form 

Here <f>^ is the velocity-potential of a motion in which q^ alone varies and 
the flux across each barrier is accordingly zero. Again Q is the velocity- 
potential of a motion in which the solids are all at rest, whilst the flux 
through the first aperture is unity, and that through every other aperture is 
zero. It is to be observed that <j>x,<f>^, . . . ^n, O, Q!, . . . are in general all of 
them cyclic functions, which may however be treated as single-valued, on the 
conventions of Art. 50. 

The kinetic energy of the fluid is given by the expression 

»-///i!i)'-(rMi)]*'*^' '^' 

where the integral is taken over the region occupied by the fluid at the 
instant under consideration. If we substitute from (1) we obtain T as a 
homogeneous quadratic function of g^, g'j, . . . j^, ;^, y^^ ;)^", . . . with coefficients 
which depend on the instantaneous configuration of the solids, and are there- 
fore functions of ji, y^, . . . ?„ oiily. Moreover, we find, by Art. 53 (1), 



aT fffidAdCl d6dCl ^ d(f>dQ) , , , 



dn '"' 

where k, k\ ... are the cyclic constants of <f), and the first surface-integral is 
to be taken over the surfaces of the solids, and the remaining ones over the 
several barriers. By the conditions which determine Q, this gives the first 
equation of the system : 

3^ = ^'^' a^'""^"' ^ ^ 

These shew that p/c, p/c', . . . are to be regarded as the generalized components 
of momentum corresponding to the velocity-components x» x'» • • • » r^p^- 
tively. 

We have recourse to the general Hamiltonian formula* (17) of Art. 135. 

* It is possible to arrange an investigation on the Lagrangian plan, parallel to that of 
Art. 136, but the proof of the formulae corresponding to (5) below involves some rather delicate 
considerations. 



139] 



Application of Hamiltonian Method 



185 



We will suppose that the varied motion of the solids is subject only to the 
condition that the initial and final configurations are to be the same as in 
the actual motion ; also that the initial position of each particle of the fluid 
is the same in the two motions. The expression 

wiU accordingly vanish at time ^q, but not in genera] at time t-i, in the 
absence of further restrictions. 

We will now suppose that the varied motion of the fluid is irrotational, 
and accordingly determined by the instantaneous values of the varied 
generalized co-ordinates and velocities. Considering the particles of the 
fluid alone, we have 

= p \\<f> (iAf + mL-q + nAO dS + pK jj{lAi + mAiy + nAO dcr 



+ PkJJ{IM + m^ri + nAO (fo' +...,. .(4) 



where I, m, n are the direction-cosines of the normal to an element of the 
bounding surface, drawn towards the fluid, or (as the case may be) of the 
normal to an element of a barrier, drawn in the direction in which the 
corresponding circulation is estimated. 

At time ti we shall have 

lAi + wAt; + nAC = 

at the surface of the solids, as well as at the fixed boundaries. Again, 

if AB represent one of the barriers in its position 

at time ti, and if A'B' represent the locus at the 

same instant, in the varied motion, of those particles 

which in the actual motion occupy the position AB, 

the volume included between AB and A'B' will be 

equal to the corresponding Ax, whence 



jj(lAi + mAri + nAO da -= Ax, 
jj(lM + mArj + nAO At' = Ax', 




(5) 



The varied circulations are, from instant to instant, still at our disposal. 
We may suppose them to be so adjusted as to make A^, A^', . . . vanish at 
time tj . The right-hand member of Art. 135 (17) will accordingly vanish, 
and if we further suppose that the extraneous forces do on the whole no 



186 Motion of Solids through a Liquid [chap, vi 

work when the boundary of the fluid is at rest, whatever relative displace- 
ments be given to the parts of the fluid, the formula reduces to 



[''{AT + QiAffi + Q^^q^ + . . . + QnAjn} * - (6) 



From this Lagrange's equations follow by a known process. We have 



Hence, by a partial integration, and remembering that by bypothesis 
A^i, Ajt, . . . iiq„, Ax, Ax', . . . vanish at the limits t^, ti, we find 

, fddT dT „ \ . , ddT . d dT . , ) ,, ^ 

(8) 

Since the values of Ajj, ^q^, ... A^n* ^X' ^x'» ••• within the range of 
integration are still arbitrary, their coefficients must separately vanish. 
We thus obtain n equations of the type 

dtdqr dqr~^" ^ ' 

togetherwith |gf = 0' llf' = ^' (^^^ 

140. Equations of the types (9) and (10) present themselves in various 
problems of ordinary Dynamics, e,g, in questions relating to gyrostats, where 
the co-ordinates x> x'» • • • > whose absolute values do not afiect the kinetic or 
the potential energy of the system, are the angular co-ordinates of the 
gyrostats relative to their frames. The general theory of such systems has 
been treated by Routh*, Thomson and Taitf, and other writers. 

* On the Stability of a Given State of Motion (Adams Prizo Essay), London, 1877 ; Advanced 
Rigid Dynamics, 6th ed., London, 1905. 

t Natural Philosophy, 2nd ed.. Art. 319 (1879). See also Helmholtz, "Principien der Statik 
monooyclischer Systeme," CreUe, t, xcyii. (1884) [Wiss. Ahh, t. iiL p. 179]; Larmor, "On the 
Direct Application of the Principle of Least Action to the Dynamics of Solid and fluid Systems," 
Proc. Lond. Math. Sog. t. xv. (1884); Lamb, Art. "Dynamics, Analytical," Encyc, Brit. 10th ed. 
t. xzvii. p. 566 (1902), 11th ed. t. viii. p. 759 (1910); Whittaker, Analytical Dynamics, c. iii 



139-141] Ignoratimi of Co-ordinates 187 

We have seen that ^-r = p#c, x-r, = pk\ .... (11) 

and the integration of (10) shews that the quantities #c, k\ . . . are constants 
with regard to the time, as is otherwise known (Art. 50). Let ns write 

R=T^Pkx-pk'x'- (12) 

The equations (11), when written in full, determine ;^, ;^', . . . as linear functions 
of /c, K,... and qi,4t» • • • ?n 5 and by substitution in (12) we can express R as 
a homogeneous quadratic fimction of the same quantities, with coefficients 
which of course in general involve the co-ordinates Ji, ?2) • • • ?n* On this 
supposition we have, performing the arbitrary variation A on both sides of 
(12), and omitting terms which cancel by (11), 

^^ A . . , 3^ A . , 3^ A , 

= g-r- A^i + . . . + g— Aji -h . . . — px^ — • • • J • • -(13) 

where, for brevity, only one term of each kind is exhibited. Hence we obtain 
2n equations of the types 

dR^dT dR^dT 

dqr a?r' 3?r 3?r ' 

together with —=^px, diP^" ^^'' ^^^^ 

Hence the equations (9) may be written 

ddR dR ^ .^ ^. 

i^d^rwr^^' ^ ^ 

where the velocities x* x'> • • • corresponding to the * ignored' co-ordinates 
Xj x'> • • • have now been eliminated*. 

141. In order to shew more explicitly the nature of the modification 
introduced by the cyclic motions into the dynamical equations, we proceed as 
follows. 

If we substitute in (12) from (15), we obtain 

r-«-('S^4*+-) ™ 

Now, remembering the composition of R, we may write for a moment 

R = -Ba,o + -Bi.i + ^,2> (18) 

where £2,0 is a homogeneous quadratic function of g^, 92? • • • 9n> -^.2 1^ & 

* This inyestigation is due to Routh, I.e. ; cf. Whittaker, Analytical Dynamics, Ait. 38. 



188 Motimi of Solids through a Liquid [chap, vi 

homogeneous quadratic function of k,k\ . . . , and Bi,i is bilinear in these two 
sets of variables. Hence (17) takes the form 

T = B,.o~Bo.«, (19) 

or, as we shall henceforth write it, 

T = ^-|-ir, (20) 

where ^ and K are homogeneous quadratic functions of g^, gj* • • • ?n> ^^'^ of 
K^K'y . . . , respectively. It follows also from (18) that 

* = ®-^-A?i-i3292-----i3«?«, (21) 

where j8, , jSj, ... are linear functions of /c, #c', . . . , say 

ft = a^K + a^K* + . . . / 
ft = aj/c + OjV + . . . , 



ft = a^K + a„V + . . . . 



(22) 



The meaning of the coefficients a (in the hydrodynamical problem) appears 
from (15) and (21). We find 






dK 

PX ^ 3;7 "^ ^1 ?> "^ ^« 9a + . . . +a„'g„, 



•/ 



/ • 



/J. 



/ • 



V 



(23) 



which shew that a, is the contribution to the flux of matter across the first 
barrier due to unit rate of variation of the co-ordinate g,., and so on. 

If we now substitute from (21) in the equations (16) we obtain the general 
equations of motion of a 'gyrostatic system,' in the form* 

diWi'Wx +(1.2)^. + (1.3)j,+ ... + (l,n)g„ + g^ 

dK 



dm m ^,^ ,.. 



+ (2,3)?,+ ... +(2,n)g, + g- = Q„ 



dm m , , ,.., , ox . i / ox . , 

*a^~a?;"^^"' ^^'"^^"' ^'*"^^"' ^^•^••' 



where 



(,, ,) = f. _ p. 

dqr dq. 



^dq„ ^"' 
(24) 

(25) 



It is important to notice that (r, s) = — («, r), and (r, r) = 0. 

* These equations were first given in a paper by Sir W. Thomson, "On the Motion of Bigid 
Solids in a Liquid circulating irrotationaliy through perforations in them or in a Fixed Solid," 
Phil Mag. May 1873 [Ptipers, t. iy. p. 101]. See also 0. Neumann, Hydrodj/namische UnUf' 
wchungen (1883). 



141-142] Equations of Motion 189 

If in the equations of motion of a fuUy-specilBled system of finite freedom 
(Art. 135 (14)) we reverse the sign of the time-element 8<, the equations are 
unaltered. The motion is therefore reversible ; that is to say, if as the system 
is passing through any assigned configuration the velocities 9i, 92, . . . 9n ^^ 
all reversed, it will (if the forces be always the same in the same configuration) 
retrace its former path. It is important to observe that this statement does 
not in general hold of a gyrostatic system ; thus, the terms in (24) which are 
linear in j^, jg? • • • ^n change sign with 8^, whilst the others do not. Hence, 
in the present application, the motion of the solids is not reversible, unless 
indeed we imagine the circulations /c, k,,,, to be reversed simultaneously 
with the velocities g,, g,, ... g„*. 

If we multiply the equations (24) hj qj, q^, . . . qn i^ order, and add, we 
find, by an obvious adaptation of the method of Art. 135, 

I (® + ^) = Q,q, + Q^q^ + . . . + (?„?„, (26) 

or, if the system be conservative, 

-21; + ^+ 7 = const (27) 

142. The results of Art. 141 may be applied to find the conditions of 
equilibrium of a system of solids surrounded by a Uquid in cyclic motion. 
This problem of * Kineto-Statics,' as it may be termed, is however more 
naturally treated by a simpler process. 

The value of <f) under the present circumstances can be expressed in the 
alternative forms 

^ = x" + x'ii'+---> (1) 

<f> = KO) + #c'a>' + . . . ; (2) 

and the kinetic energy can accordingly be obtained as a homogeneous quad- 
ratic function either of %, %', . . . , or of k,k, . . . , with coefficients which are 
in each case fimctions of the co-ordinates Ji, g'2> • • • ?n which specify the 
configuration of the solids. These two expressions for the energy may be 
distinguished by the symbols Tq and K, respectively. Again, by Art. 55 (5) 
we have a third formula 

2T = Pkx + PkY+ (3) 

The investigation at the beginning of Art. 139, shortened by the omission 
of the terms involving q^, q^, ... g„, shews that 

/>'^= a-' p^ -sY' ^ ^ 

* Just as the motion of the axis of a top cannot be reversed unless we reverse the spin. 



190 Motion of Solids through a Liquid [chap, vi 

Again, the explicit formula for K is 

■ 

= (/c, k) k^ + (k\ /c') '<^'" + . . • + 2 (/c, /c') #<r#c' + . . • , ... .(5) 



where 



(k, k) = - p\^f^ At, (k, k') = - p (J ^ At = -p\\^ ^' , ■ ■ ■ -(6) 



and so on. Hence 





dK 

dK 


(>^, 


k)k+ (k, k') 


k' + 


1 * • ^~ 


We thus obtain 






. dK 

P^=dK' 


PX = 


dK 

dK" 



(7) 



Again, writing Tq + K for 2T in (3), and performing a variation A on both 
sides of the resulting identity, we find, on omitting terms which cancel in 
virtue of (4) and (7)*, 

dqr^Wr"^ ^®^ 

This completes the requisite analytical formulae f. 

If we now imagine the solids to be guided from rest in the configuration 
{9j> 9t9 • • • 9n) to rest in an adjacent configuration 

the work required is QiAqi + OzAjg + . . . + Qn^9n9 

where Qi,Q^, ... Qn are the components of extraneous force which have to be 
applied to neutrahze the pressures of the fluid on the solids. This must 
be equal to the increment AK of the kinetic energy, calculated on the 
supposition that the circulations k, k\.,. are constant. Hence 

e, = ^ (9) 

The forces representing the pressures of the fluid on the solids (when these 
are held at rest) are obtained by reversing the signs, viz. they are given by 

e'' = -i' (10) 

the solids therefore tend to move so that the kinetic energy of the cyclic 
motion diminishes. 

* It would be sufficient to assume either (4) or (7) ; the process then leads to an independent 
proof of the other set of formulae. 

f It may be noted that the function R of Art. 140 now reduces to - ^. 



142-144] Kimto-Statics 191 

In virtue of (8) we have, also, 

«''=!; (11) 

143. A simple application of the equations (24) of Art. 141 is to the case 
of a sphere moving in a liquid which circulates irrotationally in a cyclic space 
with fixed boimdaries. 

If the radius a, say, of the sphere be small compared with its least distance from the 
fixed boundary, the formula (20) of Art. 141 becomes 

2T=m (£^ +^* +«*) +K, (1) 

where x, y, z are the co-ordinates of the centre, and m denotes the mass of the sphere to- 
gether with half that of the fluid displaced by it ; sec Art. 92. To find K, the energy of 
the cyclic motion when the sphere is held at rest in its actual position, we note that if we 
equate x, y, i to the components u, v, Wy respectively, of the fluid velocity which would 
obtain at the point (x, |/, z) if the sphere were absent, and at the same time put 
m=2n-pa^ the resulting energy will be practically the same as that of the fluid when 
filling the region, whence 

2irpa' (u^ +f^+v^) +^ =const., 

or K =const. -W (2) 

where W =^2irpa^ (w* +v* +tt'«) (3) 

Again the coefficients a^, a^, a^ of Art. 141 (22) denote the fluxes across the first barrier, 
when the sphere moves with unit velocity parallel to ar, y, z respectively. If we denote by 
O the flux across this barrier due to a unit simple-source at (x, y, z), then remembering the 
equivalence of a moving sphere to a double-source (Art. 92), we have 

ax,a„a3=K^> W ^^ i«» ^ (4) 

SO that the quantities denoted by (2, 3), (3, 1), (1, 2) in Art. 141 (24) vanish identically. 
The equations therefore reduce in the present case to 

mx=X + ^, iiiy=F + -g^, mz = Z-h-g^ (6) 

where X, F, Z are the components of extraneous force applied to the sphere. 

When X, Y, Z=0, the sphere tends to move towards places where the undisturbed 
velocity of the fluid is greatest. 

For example, in the case of cyclic motion round a fixed circular cylinder (Arts. 27, 64), 
the fluid velocity varies inversely as the distance from the axis. The sphere will therefore 
move as if under the action of a force towards this axis varying inversely as the cube of 
the distance. The projection of its path on a plane perpendicular to the axis will therefore 
be a Cotes' spiral*. 

144. We may also notice one or two problems of Kineto-Statics, in 
illustration of the theory of Art. 142. 

* Cf. Sir W. Thomson, Ic. anU p. 188. 



192 Motion of Solids through a Liquid [chap, vi 

It will be shewn in Art. 16? that the energy K of the cyclic fluid motion is propor- 
tional to the energy of a system of electric current-sheets coincident with the fixed 
boundaries, the current-lines being orthogonal to the stream-lines of the fluid. 

The electi'omagnetic forces between conductors carrying these currents are proportional * 
to the expressions on the right-hand of Art. 142 (10) with the signs reversed. Hence in 
the hydrodynamical problem the forces on the solids are opposite to those which obtain in 
the electrical analogue. In the particular case where the fixed solids reduce to infinitely 
thin cores, round which the fluid circulates, the current-sheets in question are practically 
equivalent to a system of electric currents flowing in the cores, regarded as wires, with 
strengths k, k, ... respectively. For example, two thin circular rings, having a common 
axis, will repel or attract one another according as the fluid circulates in the same or in 
opposite directions through themf. This might have been foreseen of course from the 
principle of Art. 23. 

Another interesting case is that of a number of open tubes, so narrow as not sensibly 
to impede the motion of the fluid outside them. If streams be established through the 
tubes, then as regards the external space the extremities will act as sources and sinks. 
The energy due to any distribution of positive or negative sources my^m^, ... is given, so 
far as it depends on the relative configuration of these, by the integral 



-\p\\<t>^da, (6) 



taken over a system of small closed surfaces surrounding m^, m,, ... respectively. If 
<f>i, <t>t, ... be the velocity-potentials due to the several sources, the part of this expression 
which is due to the simultaneous presence of 914, m, is 

-*''//(^l*-*.^')'^' (') 

which is by Green's Theorem equal to 



-f>jj<t>^ 



'-^^^- (8) 



Since the surface-integral of d<f>Jdn is zero over each of the closed surfaces except the 
one surrounding m^, we may ultimately confine the integration to this, and so obtain 



"''**// "^^^>=^«** (^) 



Since the value of <^| at m, is mi/^irfi^, where r,, denotes the distance between m^ and m,, 
we obtain, for the part of the kinetic energy which varies with the relative positions of the 
sources, the expression 

h'T: ^''^ 

The quantities m^, m^, ... are in the present problem equal to the fluxes xo» Xo^ • • • 
across the sections of the respective tubes, so that (10) corresponds to the form Tg of the 

* Maxwell, Electricity and Magnetism, Art. 573. 

t Tho theorem of this paragraph was given by Kirchhoff, l.c. ante p. 62. See also Sir W. 
Thomson, "On the Forces experienced by Solids immersed in a Moving Liquid," Proc. R, S. 
Edin. 1870 [Reprint, Art. xli.]; Boltzmami, **Ueber die Druckkr&fte welche auf Ringe wirksam 
Bind die in bewegte Flussigkeit tauchen," CreUe, t. Ixxiii (1871) [Wise. Abh, t. L p. 200.] 



144] Kineto-StaticB 193 

kinetio energy. The force apparently exerted by m^ on m,, tending to increase fn, is 
therefore, by Art. 142 (11), 

Hence two sources of like sign attract, and two of unlike sign repel, with forces varying 
inversely as the square of the distance*. This result, again, is easily seen to be in accord- 
ance with general principles. It also follows independently from the electric analogy, the 
tubes corresponding to Ampdre's * solenoids.' 

We here take leave of this branch of our subject. To avoid, as far as 
may be, the suspicion of vagueness which sometimes attaches to the use of 
'generalized co-ordinates/ an attempt has been made in this Chapter to put 
the question on as definite a basis as possible, even at the expense of some 
degree of prolixity in the methods. 

To some writers f the matter has presented itself as a much simpler one. 
The problems are brought at one stroke imder the sway of the ordinary 
formulae of Dynamics by the imagined introduction of an infinite number of 
'ignored co-ordinates,' which would specify the configuration of the various 
particles of the fluid. The corresponding components of momentum are 
assumed all to vanish, with the exception (in the case of a cyclic region) 
of those which are represented by the circulations through the several 
apertures. 

Prom a physical point of view it is difficult to refuse assent to such 
a generalization, especially when it has formed the starting-point of all the 
development of this part of the subject; but it is at least legitimate, and 
from the hydrodynamical standpoint even desirable, that it should be 
verified a posteriori by independent, if more pedestrian, methods. 

Whichever procedure be accepted, the result is that the systems con- 
templated in this Chapter are found to comport themselves (so far as the 
'palpable' co-ordinates ft, ft* ••• ?n *r® concerned) exactly Uke ordinary 
systems of finite freedom. The further development of the general theory 
belongs to Analytical Dynamics, and must accordingly be sought for in books 
and memoirs devoted to that subject. It may be worth while, however, to 
remark that the hydrodynamical systems afEord extremely interesting and 
beautiful illustrations of the Principle of Least Action, the Reciprocal 
Theorems of Helmholtz, and other general dynamical theories. 

* Sir W. Thomaon, Ic 

t See Thomson and Tait, and Larmor, U. ciL ante p. 186. 



L. H. 13 



CHAPTER VII 

VORTEX MOTION 

145. Our investigations have thus far been confined for the most part 
to the case of irrotational motion. We now proceed to the study of 
rotational or 'vortex' motion. This subject was first investigated by 
Hehnholtz*; other and simpler proofs of some of his theorems were after- 
wards given by Kelvin in the paper on vortex motion abeady cited in 
Chapter in. 

We shall, throughout this Chapter, use the symbols f, ly, ^ to denote, as 
in Chapter in., the components of vorticity, viz. 

^~a^"a^' '^"di'^di' ^"a5""a^ ^^ 

A line drawn from point to point so that its direction is everywhere 
that of the instantaneous axis of rotation of the fluid is called a 'vortex-line.' 
The differential equations of the system of vortex-lines are 

dx dv dz ,^, 

-y = ^ = — (2) 

? V C 

If through every point of a small closed curve we draw the corresponding 
vortex-line, we mark out a tube, which we call a * vortex-tube.' The fluid 
contained within such a tube constitutes what is called a 'vortex-filament,' 
or simply a 'vortex.' 

Let ABC, A'B'C be any two circuits drawn on the surface of a vortex- 
tube and embracing it, and let AA' be a connecting line 
also drawn on the surface. Let us apply the theorem 
of Art. 32 to the circuit ABCAA'C'B'A'A and the part 
of the surface of the tube bounded by it. Since 

at every point of this surface, the Une-integral 

/ {udx + vdy + wdz)y 

* "Ueber Integrale der hydrodynamischen Gleiohungen welche den Wirbelbewegungen 
enteprechen," CreOe, t. Iv. (1858) [Wiss, Ahh. t. i. p. 101]. 




145] VorteooFUaments 195 

taken round the circuit, must vanish ; ».e. in the notation of Art. 31 

I (ABC A) + I {AA') + / {A'C'B'A') + I (A' A) = 0, 

which reduces to / (ABCA) = I {A'B'C'A'), 

Hence the circulation is the same in all circuits embracing the same vortex- 
tube. 

Again, it appears from Art. 31 that the circulation round the boundary 
of any cross-section of the tube, made normal to its length, is axr, where 

<«>> = (^ + ^* + i*) > is the resultant vorticity of the fluid, and a the infinitely 
small area of the section. 

Combining these results we see that the product of the vorticity into the 
cross-section is the same at all points of a vortex. This product is conveniently 
taken as a measure of the 'strength' of the vortex*. 

The foregoing proof is due to Kelvin ; the theorem itself was first given 
by Helmholtz, as a deduction from the relation 

i+l+i-» p' 

which follows at once from the values of ^, ly, t, given by (1). In fact writing, 
in Art. 42 (1), |^, iq, t, iot J7, F, Tf , respectively, we find 

lim^-mri-VnDdS^O, (4) 

where the integration extends over any closed surface lying wholly in the 
fluid. Applying this to the closed surface formed by two cross-sections of a 
vortex-tube and the part of the walls intercepted between them, we find 
cu^oT] =co2<72, where cu^, 0)2 denote the vorticities at the sections or^, c72, 
respectively. 

Kelvin's proof shews that the theorem is true even when ^, ly, t, are 
discontinuous (in which case there may be an abrupt bend at some point of a 
vortex), provided only that w, t), w are continuous. 

An important consequence of the above theorem is that a vortex-line 
cannot begin or end at any point in the interior of the fluid. Any vortex- 
lines which exist must either form closed curves, or else traverse the fluid, 
beginning and ending on its boundaries. Compare Art. 36. 

The theorem of Art. 32 (3) may now be enunciated as follows : The 
^ circulation in any circuit is equal to the sum of the strengths of all the 
vortices which it embraces. 

* The circvlaiion round a vortex being the most natural measure of its intensity. 

la— 2 



</./. 



196 Vortex Motion [ch. vn 

146. It was proved in Art. 33 that in a perfect fluid whose' density 
is either uniform or a function of the pressure only, and which is subject 
to forces having a single-valued potential, the circulation in any circuit 
moving with the fluid is constant. 

Applying this theorem to a circuit embracing a vortex-tube we find that 
the strength of any vortex is constant. 

If we take at any instant a surface composed wholly of vortex-lines^ 
the circulation in any circuit drawn on it is zero, by Art. 32, for we have 
1$ -{- mrj + n^ = 8kt every point of the surface. The preceding Art. shews 
that if the surface be now supposed to move with the fluid, the circulation 
will always be zero in any circuit drawn on it, and therefore the surface will 
always consist of vortex-lines. Again, considering two such surfaces, it is 
plain that their intersection must always be a vortex-Une, whence we derive 
the theorem that the vortex-Unes move with the fluid. 

This remarkable theorem was first given by Helmholtz for the case of 
incompressibility ; the preceding proof, by Kelvin, shews that it holds for 
all fluids subject to the conditions above stated. 

The theorem that the circulation in any circuit moving with the fluid ia 
^ -^ invariable constitutes t he sole and sufficient ft ppeal to Dynami cs which it 
is necessary to make in the investigations of this Chapter^ It is based on 
the hypothesis of a continuous distribution of pressure, and (conversely) 
implies this. For if in any problem we have discovered functions u, v, w of 
X, y, z, t, which satisfy the kinematical conditions, then, if this solution is 
to be also dynamically possible, the relation of the pressures about two 
moving particles A^ B must be given by the formula (2) of Art. 33, viz. 



B D fB 



j??+Q^jj2 =-gj {udx-\-vdy + wdz) (1) 



It is therefore necessary and sufficient that the expression on the right hand 
should be the same for all paths of integration (moving with the fluid) which 
can be drawn from A to B. This is secured if, and only if, the assumed 
values of u, v, w make the vortex-lines move with the fluid, and also make 
the strength of every vortex constant with respect to the time. 

It is easily seen that the argument is in no way impaired if the assumed 
values of u, v, w make ^, 77, ^ discontinuous at certain surfaces, provided 
only that w, v, w are themselves everywhere continuous. 

On account of their historical interest, one or two independent proofs of the preceding^ 
theorems may be briefly indicated, and their mutual relations pointed out. 

Of these proofe, perhaps the most conclusive is based upon a slight generalization of 
some equations given originally by Cauchy in the introduction to his great memoir on 
Waves*, and employed by him to demonstrate Lagrange's velocity-potential theorem. 

* 2.C. ante p. 10. 



146] 



Persistence of Vortices 



1»7 



The equatioiia (2) of Art. 16 yield, on elimination of the function x ^7 crose-differentia- 
tion* 

8u^ du^ 'bvhy dvh/ dwdz dw dz ^ dw^ dv^ 
dbde' dc db 569c dcdb db dc^ ^ db~ ~^ ~ ^ 

(where u, v, to have been written in place of dx/dt, dy/dt, dz/dt, respectively), with two 
symmetrical equations. .If in these equations we replace the differential coefficients of 
Uf V, to with respect to a, &, c, by their values in terms of differential coefficients of the 
same quantities with respect to x, y, z, we obtain 

^d{y,z) .d{z,x) , .9(a?,y)_> \ 

^a(6,c)'*"*'a(6,c)'*"^a(fe, c)"^«' 

^d(a, b) ^^d{a, 6) "*"' 8(a, b) ""'*'• ; 

If we multiply these by dx/da, dz/db, dx/dCy in order, and add, then, taking account of 
the Lagrangian equation of continuity (Art. 14 (1)) we deduce the first of the following 
three symmetrical equations: * 

p p^da po 96 podc' 

^=:^^+5?^+(5?y, y (3) 

p Po^ Po^ Po^' ' 

p Po ^ Po ^ Po ^ 



In the particular case of an incompressible fluid (p =po) these differ only in the use of 
the notation f , rj, ( from the equations given by Cauchy. They shew at once that if the 
initial values fo' '7o> Co ^^ ^^® component vorticities vanish for any particle of the fluid, then 
(, 17, { are always zero for that particle. This constitutes in fact Cauchy's proof of Lagrange's 
theorem. 

To interpret (3) in the general case, let us take at time ^ =0 a linear element coincident 
with a vortex-line, say 

Po Po Po 

where c is infinitesimal If we suppose this element to move with the fluid, the equations 
(3) shew that its projections on the co-ordinate axes at any other time will be given by 

asr, «y, &=€^, f^. €^, 
P P P 

ue, the element will still form part of a yortex-line, and its length (df, say) will vary as 
a>/p, where » is the resultant vortioity. But if a- be the cross-section of a vortex-filament 
having ds as axis, the product pads is constant with regard to the time. Hence the strength 
wr of the vortex is constant*. 



* See NaoBon, Mega, of Maih. t. iiL p. 120 (1874); Kirchhoff, Meckanik, c. xv. (1876); 
Stokes, Papers, t. ii p. 47 (1883). 



198 



Vortex Motion 



[oh. vn 



The proof given originally by Helmholtz depends on a system of three equations 
which, when generalized so as to apply to any fluid in which p is a function of p only, 
become* 



Dt\p) p dx pdy p^' 

Dt \pj pdx pdy p^' 

R(i\ = i^ 1^ ^^ 

Dt \pj "pdx pdy p dz' J 



(4) 



These may be obtained as follows. The dynamical equations of Art. 6 may be written, 
when a force-potential O exists, in the forms 



provided 



du ^ 8y' \ 

dw ^ dv' 



(6) 



(6) 



where 9* =u* +t)* +(«*. From the eeoond and third of these we obtain, «lit»ina.t ing ^' by 
oroea-diSerentiation, 

3* d( 31 /a> d{\ du ,du ,/dv dtv\ 



Remembering the relation 




dx 


4-^ + ^f- 

^ dz 


=0, 


and the equation of continuity 














Dp 
Dt 


^p{ 


fdu 
idx 


dv 
dy"- 


dm 

dz 



(7) 



we easily deduce the first of equations (4). 

To interpret these equations we take, at time t, a linear element whose projections on 
the co-ordinate axes are 



^ n C 
P P P 



(9) 



where c is infinitesimaL If this element be supposed to move with the fluid, the rate at 
which bx is increasing is equal to the difference of the values of « at the two ends, whence 



Dbx __ ^ du Tj du ( du 
Dt p dx p dy p d^' 



It follows, by (4), that 



5(*^-'f)=«' s(*»'-'j)=«-' s(»*-'^)=« <i«> 



* Nanson, l.c. 



146] HdmhoUz' Equatimis 199 

Helmholtz oonoludee that if the relations (9) hold at time t^ they will hold at time 
t-\-btt and so on, continually. The inference is, however, not quite rigorous; it is in fact 
open to the criticisms which Stokes* directed against various defective proofs of Lagrange's 
velocity-potential theorem f. 

By way of establishing a connection with Kelvin's investigation we may notice that 
the equations (2) express that the circulation is constant in each of three infinitety small 
circuits initially perpendicular, respectively, to the three co-ordinate axes. Taking, for 
example, the circuit which initially bounded the rectangle d6 dc, and denoting by A, B, C 
the areas of its projections at time t on the co-ordinate planes, we have 

9 (6, C) d{b,c) (6, c) 

SO that the first of the equations referred to is equivalent :( to 

^A +ffB + {C =&a6dc (11) 

As an application of the equations (4) we may consider the motion of a liquid of uniform 
vorticity contained in a fixed ellipsoidal vessel§. The formulae 

u—qz-ry, v^^rx-pZy w=py-qx (12) 

obviously represent a uniform rotation of the fluid as a solid within a spherical boundary. 
Transforming the co-ordinates and the corresponding velocities by homogeneous strain we 
obtain the formulae 

u _qz ry v _rx pz w _py gx ,,ox 

« ~ ~r jT* t "~ ~z zr » i ~~ t. it* ••••••••••••• •k*"^ 

acooaccoa 

as representing a certain motion within a fixed ellipsoidal boundary 

^+^+^=1- ••• (1*) 

Substitotiiig in (4) we obtain 

(6»+c«)J={6«-c«)gr. (16) 

* Ic, ante p. 16. 

t It may be mentioned that, in the case of an incompressible fluid, equations somewhat 
similar to (4) had been established by Lagrange, MisctU, Taur. t. ix. (1760) [Oeuvrea, t. i. p. 442]. 
The author is indebted for this reference, and for the above remark on Helmholtz* investi- 
gation, to Sir J. Larmor. Equations equivalent to those given by Lagrange were obtained 
independently by Stokes, Ix. and made the basis of a rigorous proof of the velocity-potential 
theorem. 

% Nanson, Meae, of Math. t. viL p. 182 (1878). A similar interpretation of Helmholtz* 
equations was given by the author of this work in the Meas, of Math, t. vii. p. 41 (1877). 

Finally it may be noted that another proof of Lagrange's theorem, based on elementary 
dynamical principles, without special reference to the hydrokinetic equations, was indicated by 
Stokes, Comb, Trans, t, viii. [Papere, t. i p. 113], and carried out by Kelvin in his paper on 
Vortex Motion. 

S Cf. Voigt, "Beitrftge zur Hydrodynamik,'* Gott, Nachr, 1891, p. 71 ; Tedone, Nuovo CimetUo, 
t, xxxiiL (1893). The artifice in the text is taken from Poincar6, "Sur la precession dee corps 
d^ormables,*' BvU. Asir. 1910. 



200 Vortex Motion [oh. vn 

whioh may be written 

a«(6« +c«) ^ = {6«(c«+a*) -c*(a*+6*)} qr, (17) 

with two similar equatiouB. We have here an identity as to fonn with Euler's equations 
of free motion of a solid about a fixed point. We easily deduce the integrals 

^ + ?1 + ^ = const. (18) 

a* b^ c* ^ ' 

6Vf* c«o«i;» a«6«f« 

and iT— H+-i — ~i+ • M =oonst, (19) 

6"+c* e*+a* a*+b* ^ ' 

the former of which is a verification of one of Heknholtz' theorems, whilst the latter follows 
from the constancy of the energy. 

147. It is easily seen by the same kind of argument as in Art. 41 that 
no continuous irrotational motion is possible in an incompressible fluid filling 
infinite space, and subject to the condition that the velocity vanishes at 
infinity. This leads at once to the following theorem : 

The motion of a fluid which fills infinite space, and is at rest at infinity, 
is determinate when we know the values of the expansion (0, say) and of the 
component vorticities ^, 77, ^, at all points of the region. 

For, if possible, let there be two sets of values, u^, Vi, Wi, and u^y v^y w^, 
of the component velocities, each satisfjdng the equations 

dx^dy^ dz ^' ^^^ 

throughout infinite space, and vanishing at infinity. The quantities 

m' = Wj — Wjj v' = V^ — Vj, w' = M^i — W2 

will satisfy (1) and (2) with 6, f , >y, J = 0, and will vanish at infinity. Hence, 
in virtue of the result above stated, they will everywhere vanish, and there 
is only one possible motion satisfying the given conditions. 

In the same way we can shew that the motion of a fluid occupying any 
limited simply-connected region is determinate when we know the values of 
the expansion, and of the component vorticities, at every point of the region, 
and the value of the normal velocity at every point of the boundary. In the 
case of an n-ply-connected region we must add to the above data the values 
of the circulations in n several independent circuits of the region. 



146-148] Velocities due to a Vortex-System 201 

148. If, in the case of infinite space, the quantities 0, f , t;, ^ all vanish 
beyond some finite distance of the origin, the complete determination of 
UyVfW'wi terms of them can be effected as follows*. « 

The component velocities due to the expansion can be written down at 
once from Art. 56 (1), it being evident that the expansion 0' in an element 
Sx'Sy'Sz' is equivalent to a simple source of strength 0'Sx'8y'8z\ We thus 
obtain 

9^ 34> 9^ ... 

^"■"^' ""^"d^' ^^^^' ^^) 

where ' ^^^ fjjjdafdy'dz^, (2) 

r denoting the distance between the point {x'y y\ z') at which the volume- 
element of the integral is situate and the point {x, y, z) at which the values 
of u, V, w are required, viz. 

f = {(a? - xr + {y- y'Y + (2 - «')*}*, 

and the integration including all parts of space at which ff differs from zero. 

To find the velocities due to the vortices^ we note that when there is no 
expansion, the flux across any two open surfaces bounded by the same curve 
as edge will be the same, and will therefore be determined solely by the 
configuration of the edge. This suggests that the flux through any closed 
curve may be expressed as a line-integral taken round the curve, say 

i(Fdx + Ody + Hdz) (3) 

On this hypothesis we shall have, by the method of Art. 31, 

_dH dO dFdH _dO dF 

^""9^ 9z' ^^dz dx' ^"^dx dy ^*^ 

To test the assumption, we must have 

9t«^_9v_ 9_/aF; 9^ 9ff\ 
^~9y dz'^dxKdx'^ dy^ dz) ^' 

with two similar equations. The quantities F, 0, H will in any case be 
indeterminate to the extent of three additive functions of the forms 
dx/dxy dx/dyy 9^/92;, respectively; and we may, if we please, suppose x to 
be chosen so that 

di + d^-^W^' (^) 

in which case ^*F = - ^, V«(?=-ij, V»^ = -J (6) 

* The investigation which follows is substantially that given by Heimholtz. The kine- 
matical problem in qnestion was first solved, in a slightly different manner, by Stokes, "On 
the Dynamical Theory of Diffraction," Camb. Trant. t. ix. (1849) [Paftrn, t iL pp. 2M. . .]. 



202 Vortex Motion [oh. vn 

Paxticular solutions of these equations are obtained b;^ equating F, Gy H to 
the potentials of distributions of matter whose volume-densities are f/^Tr, 
>y/47r, C/^TT* respectively ; thus 

^-^JIJ^T^'^tf'dz', O^ljlj^^dx'dy'dz', H = ljjj^;d.'d/dz', 

(7) 

where the accents attached to |, 17, ^ are used to distinguish the values of 
these quantities at the point (x\ y\ %'). The integrations are to include, 
of course, all places where ^, 17, J difEer from zero. ' 

Moreover, since 3/9a? . f-i s= — 3/3x' . r-^, the formulae (7) make 

The right-hand member vanishes, by the theorem of Art. 42 (4), since 

dx dy dz 

everywhere, whilst If H- m>y H- nj = 

at the surfaces of the vortices (where f, 17, ^ may be discontinuous), and 
ii Vy ^ vanish at infinity. Hence no additions to the values (7) of F, G, H 
are necessary in order that (5) may be satisfied. 

The complete solution of our problem is obtained by superposition of the 
results contained in (1) and (4), viz. 



dx dy dz ' 

dy dz dx 
34) . dG dF 



(8) 



325 dx dy ' 

where <!>, F, G, H have the values given in (2) and (7). 

It may be added that the proviso that d, f , r), J should vanish beyond a 
certain distance from the origin is not absolutely essential. It is sufficient 
if the data be such that the integrals in (2) and (7), when taken over infinite 
space, are convergent. This will certainly be the case if fl, f, ly, ^ are 
ultimately of the order jB-»», where R denotes distance from the origin, and 
n>3*. 

When the region occupied by the fluid is not unlimited, but is bounded 
(in whole or in part) by surfaces at which the normal velocity is given, and 

♦ Cf. Leathern, Cambridge TracU, No. 1 (2nd ed.), p. 44. 



148-149] Electro-magnetic Analogy 203 

when further (in the case of an n-ply connected region) the value of the 
circulation in each of n independent circuits is prescribed, the problem may 
by a similar analysis be reduced to one of irrotational motion, of the kind 
considered in Chapter in., and there proved to be determinate. This may be 
left to the reader, with the remark that if the vortices traverse the region, 
beginning and ending on the boimdary, it is convenient to imagine them 
continued beyond it, or along the boundary, in such a manner that they form 
re-entrant filaments, and to make the integrals (7) refer to the complete 
system of vortices thus obtained. On this imderstanding the condition (5) 
will still be satisfied. 

There is an exact correspondence between the analytical relations above developed and 
those which obtain in the theory of Electro-magnetism. If, in the equations (1) and (2) 
of Art. 147, we write 

a, ft y, p, tt, v, w, p 

for u, V, w, B, f , 17, Cf ^f 

respectively, we obtain 



9a 3/3 9y __ 
dx ^ dz ""'^' 



(9) 



9y 9/3 __ 9a 9y _ 9j3 9a __ I 

9y dz~ ' dz S~ ' ^ dy~ *) 

which are the fundamental relations of the theory referred to; viz. a, /3, y are the oolnpo- 
nents of magnetic force, «, v, U7 those of electric current, and p is the volume-density of the 
imaginary magnetic matter by which any magnetization present in the field may be repre- 
sented*. Hence, the vortex-filaments correspond to electric circuits, the strengths of the 
vortices to the strengths of the currents in these circuits, sources and sinks to positive and 
negative magnetic poles, and, finally, fluid velocity to magnetic force f. 

The analogy will of course extend to all results deduced from the fundamental relations ; 
thus, in equations (8), 4 corresponds to the magnetic potential and F,0,HU> the com- 
ponents of 'electro-magnetic momentum.' 

149. To interpret the result contained in Art. 148 (8), we may calculate 
the values of u, v, w due to an isolated re-entrant vortex-filament situate in an 
infinite mass of incompressible fluid which is at rest at infinity. 

Since = 0, we shall have <E> = 0. Again, to calculate the values of 
Fy Gj Hy we may replace the volume-element Sx'Sy'Sz' by a'S*', where S*' is 
an element of the length of the filament, and a' its cross-section. Also 

f=a.^, ^=0.5^-., C=a.^. 

* Cf. Maxwell, SledrieUy and MagneHsm, Art. 607. The comparison has been simplified by 
the adoption of the 'rational* system of electrical units advocated by Heaviside, Eleetrical Papers, 
London, 1892, 1. 1 p. 199. 

t This analogy was first pointed out by Helmholtz; it has been extensively utilized by 
Kelvin in his papers on BleclraHaties and Magnetism (cited ante p. 37). 



204 



Vortex Motion 



[oh. vn 



frhere at' is the Torticity. Hence the formulae (7) of Art. 148 become 

^-iirJT' ^-^]r' '^ 4,7.1 f' ^^' 

where #c, = co V, measures the strength of the vortex, and the integrals are 
to be taken along the whole length of the filament. 



Hence, by Art. 148 (4), we have 



wz\-*y')' 



with similar results for v, w. We thus find* 



_ K^ t/dy^ z- z' _ d^ y-^ \ ds^ ' 
**~4»rjl(fo' r ds' r )r*' 

— — {(^^' '" ~^' _ ^ g — g' \ <fe' 
^~4^)\ds' r ds' r Jr>'} 



(2) 



If Aw, Av, ^w denote the parts of these expressions which involve the 
element ha' of the filament, it appears that the resultant of At^, Av, ^w is 
perpendicular to the plane containing the direction of the vortex-line at 
(x\ y\ z') and the line r, and that its sense is that in which the point (a?, y, z) 
would be carried if it were attached to a rigid body rotating with the fluid 
element at (a?', y', z'). For the magnitude of the resultant we have 



{(A«)« + (^vy + (A«;)«}» = ^ ^,^'. 



(3) 



where x is the angle which r makes with the vortex-line at {x\ y\ z'). 



With the change of symbols indicated in the preceding Art. this result becomes identical 
with the law of action of an electric current on a magnetic polef. 



Velocity -PoterUial due to a Vortex. 

150. At points external to the vortices there exists a velocity-potential, 
whose value may be obtained as follows. Taking for shortness the case of a 
single re-entrant vortex, we have from the preceding Art., in the case of an 
incompressible fluid. 



«-£/(l';-*'-ii^') m 



* These are equivalent to the forms obtained by Stokes, Uc, arUe p. 201. 

t Ampere, TMorie mathinuUigue des phinomines ^ectro-dynamiquea, Paris, 1826. 



149-150] Velocity-Potential due to a Vortex 205 

By Stokes' Theorem (Art. 32 (4)) we can replace a line-integral extending 
round a closed curve by a surface-integral taken over any surface bounded 
by that curve ; viz. we have, with a slight change of notation, 

/,.^.w..^)=//{.(^-g)^».(g-g)-(g-|))^'. 

Bweput P-O, 0-1 !, Jt--^\. 

we find 

%' dz' dx'^r" dz' dx' dx'dy' r" dx' dy'~dx'dz'r" 
so that (1) may be written 

Hence, and by similar reasoning, we have, since d/dx' . r-^ = — d/dx . f-*, 

"=-S' ^=-^' «'="!' (2) 

where ^ = ^JJ(,|, + ,| + ,|,)1 ^' (3) 

Here I, w, n denote the direction-cosines of the normal to the element SS' of 
a surface bounded by the vortex-filament. 

The formula (3) may be otherwise written 



*-s(T 



""JdS' (4) 



where ^ denotes the angle between r and the normal (Z, m, n). Since 
cos 6 SS'jr* measures the elementary solid angle subtended by SS' at (Xy y, z), 
we see that the velocity-potential at any point, due to a single re-entrant 
vortex, is equal to the product of K/4frr into the solid angle which a 
surface boimded by the vortex subtends at that point. 

Since this soUd angle changes by ^tt when the point in question describes 
a circuit embracing the vortex, we verify that the value of <f> given by (4) is 
cyclic, the cyclic constant being k. Cf. Art. 145. 

It may be noticed that the expression in (4) is equal to the flux (in the 
negative direction) through the aperture of the vortex, due to a point-source 
of strength k at the point {x, y, z). 

Comparing (4) with Art. 66 (4) we see that a vortex is, in a sense, 
equivalent to a uniform distribution of double sources over any surface 
bounded by it. The axes of the double sources must be supposed to be 



206 Vortex Motion [ch. vn 

everywhere normal to the surface, and the density of the distribution to be 
equal to the strength of the vortex. It is here assumed that the relation 
between the positive direction of the normal and the positive direction of 
the axis of the vortex-filament is of the 'right-handed' type. See Art. 31. 

Conversely, it may be shewn that any distribution of double sources over 
a closed surface, the axes being directed along the normals, may be replaced 
by a system of closed vortex-filaments lying in the surface*. The same thing 
will appear independently from the investigation of the next Art. 

Vortex-Sheets. 

151. We have so far assumed w, v, w to be continuous. We may now 
shew how cases where surfaces of discontinuity present themselves may be 
brought within the scope of our theorems. 

The case of a discontinuity in the normal velocity alone has already 

been treated in Art. 58. If w, v, w denote the component velocities on one 

side, and u\ v\ w' those on the other, it was found that the circumstances 

could be represented by imagining a distribution of simple sources, with 

surface-density 

l{u' ~ u) + m (v' — v) + n(w' — w), 

where {, m, n denote the direction-cosines of the normal drawn towards the 
side to which the accents refer. 

Let us next consider the case where the tangential velocity (only) is 
discontinuous, so that 

l(u' -u) + m{v' '-v)-\-n{w' -w) = (1) 

We will suppose that the lines of relative motion, which are defined by the 

differential equations 

dx dv dz 
= -^ =-7^^—, (2) 

are traced on the surface, and that the system of orthogonal trajectories to 
these lines is also drawn. Let PQ, P'Q' be linear elements drawn close to 
the surface, on the two sides, parallel to a line of the system (2), and let PP' 
and QQ' be normal to the surface and infinitely small in comparison with PQ 
or P'Q', The circulation in the circuit P'Q'QP will then be equal to 
(?' ~" i) PQi where q, q' denote the absolute velocities on the two sides. This 
is the same as if the position of the surface were occupied by an infinitely 
thin stratum of vortices, the orthogonal trajectories above-mentioned being 
the vortex-lines, and the vorticity cj and the (variable) thickness Sn of the 
stratum being connected by the relation 

(o8n =z q' -- q (3) 

* Of. Maxwell, Electricity and Magnetism, Arts. 486, 652. 



150-151] Vortex-Sheets 207 

The same result follows from a consideration of the discontinuities which 
occur in the values of u, v, w as determined by the formulae (4) and (7) of 
Art. 148, when we apply these to the case of a stratum of thickness 8n within 
which ^, Tjy £ are infinite, but so that ^8n, lySw, $8n are finite*. 

It was shewn in Arts. 147, 148 that any continuous motion of a fluid 
filling infinite space, and at rest at infinity, may be regarded as due to 
a suitable arrangement of sources and vortices distributed with finite density. 
We have now seen how by considerations of continuity we can pass to the 
case where the sources and vortices are distributed with infinite volume- 
density, but finite surface-density, over surfaces. In particular, we may take 
the case where the infinite fluid in question is incompressible, and is divided 
into two portions by a closed surface over which the normal velocity is 
continuous, but the tangential velocity discontinuous, as in Art. 68 (12). 
This is equivalent to a vortex-sheet; and we infer that every continuous 
irrotational motion, whether cyclic or not, of an incompressible substance 
occupying any region whatever, may be regarded as due to a certain 
distribution of vortices over the boundaries which separate it from the rest 
of infinite space. In the case of a region extending to infinity, the distri- 
bution is confined to the finite portion of the boundary, provided the fluid be 
at rest at infinity. 

This theorem is complemelitary to the results obtained in Art. 58. 

The foregoing conclusions may be illustrated by means of the results of Art. 01. Thus 
when a normal velocity 8^ was prescribed over the sphere r=a, the values of the velocity- 
potential for the internal and external space were found to be 



a /r\* « , . a /a^^'*-^ 



respectively. Hence if d« be the angle which a linear element drawn on the surface 
subtends at the centre, the relative velocity estimated in the direction of this element 
will be 

2n + l d8^ 



n(n + l) 8c ' 

The resultant relative velocity is therefore tangential to the surface, and perpendicular to 
the contour lines {8f^=coDst.) of the surface-harmonic 8^, which are therefore the vortex- 
lines. 

For example, if we have a thin spherical shell filled with and surrounded by liquid, 
moving as in Art. 92 parallel to the axis of x, the motion of the fluid, whether internal or 
external, will be that due to a system of vortices arranged in parallel circles on the sphere ; 
the strength of an elementary vortex being proportional to the projection, on the axis of x, 
of the breadth of the corresponding zone of the surfacef. 

* Helmholtz, l,c. ante p. 194. 

t The same statements hold also for an ellipsoidal shell moving parallel to one of its 
principal axes. See Art. 114. 



208 Vortex Motion [oh. vn 



Impulse and Energy of a Vortex-System. 

152. The foUowmg investigations relate to the case of a vortex-system 
of finite dimensions in an incompressible fluid which fills infinite space and 
is at rest at infinity. 

The problem of finding a distribution of impulsive force (X\ Y\ Z') per 
unit mass which would generate the actual motion {u, v, w) instantaneously 
from rest is to some extent indeterminate, but a sufficient solution for our 
purpose may be obtained as follows. 

We imagine a simply-connected surface /S to be drawn enclosing all the 
vortices. We denote by <f> the single- valued velocity-potential which obtains 
outside Sy and by <f>i that solution of V^^ = which is finite throughout the 
interior of S, and is continuous with <f> at this surface.- In other words, (f>i is 
the velocity-potential of the motion which would be produced within S by 
the application of impulsive pressures fxf} over the surface. If we now assume 

X' = u + ^-K Y' = v + ^\ Z' = w + ^' (1) 

OX oy' dz ' 

at internal points, and 

z' = o, r' = o, z' = o : (2) 

at external points, it is evident on reference to Art. 11 that these forces would 
in fact generate the actual motion instantaneously from rest, the distribution 
of impulsive pressure being given by p<f> at external, and p<f>i at internal, points. 
The forces are discontinuous at the surface, but the discontinuity is only in 
the normal component, the tangential components vanishing just inside and 
just outside owing to the continuity of (f> with (f>i. Hence if (I, m, n) be the 
direction-cosines of the inward normal, we shall have 

mZ' -wr = 0, nZ'-JZ' = 0, lY'-mX' = 0, (3) 

at points just inside the surface. 

Now if we integrate over the volume enclosed by S we have 
jjjiyC - «?) ^dydz = jjj |y (I - I) - z g - g)]. dxdydz 



-/f/H^'-f)-'(f-f)l"»* 



■= - /J{y (ly -mX')-z {nX' - IZ')} dS+2 ii^X' dxdydz, ... (4) 
where the suiface-integral vanishfis in virtue of (3). 



152-153] Impulse of a Vortex-System 209 

Again 

- ///(y* + ^») idxdydz^- jjfiy* + z«) (^ - I) dxdydz 



= -//^'-^^'Hf-^>'^^^^ 



..(6) 



= my* + «•) (mZ'-nY')dS + 2 ///(yZ' - zY') dxiyiz, .... (5) 
where the surface-integral vanishes as before. 

We thus obtain for the force- and couple-resultants of the impulse of the 
Tortex-system the expressions 

^ = iP iiSiyC -zri) dxdydz, i = - i/o ///(y* + 2«) $ dxdydz, 
Q = i/» iiSi^ - xO dxdydz, M = - i/) /;;(z» -I- X*) 7) dxdydz, 
R^'Ip ///(as? - yi) dxdydz, N = - ^p //J(a;» -|- y«) J <irdy dz. J 

To apply these to the case of a single re-entrant vortex-filament of infinitely 
small section a, we replace the volume element by oSs, and write 

J. dx dy y dz 

$ = o>-^, V = <^^, C=co^ (7) 

Hence P = ipcjo i(ydz — zdy) = Kp JJWS', (8) 

L = -ipaHTS(y^'\-z^)dx==''KpSS{mz^ny)d8\ (9) 

with similar formulae. The line-integrals are supposed to be taken along the 
filament, and the surface-integrals over a barrier bounded by it. The 
identities of the different forms follow from Stokes' Theorem. We have also 
written k for okt, i.e. k is the circulation round the filament*. 

The whole investigation has reference of course to the instantaneous state 
of the system, but it may be recalled that, when no extraneous forces act, 
the impulse is, by the argument of Art. 119, constant in every respect. 

153. Let us next consider the energy of the vortex-system. It is easily 

proved that under the circumstances presupposed, and in the absence of 

extraneous forces, this energy will be constant. For if T be the energy 

of the fluid bounded by any closed surface S, we have, putting F = in 

Art. 10 (5), 

DT 

-jT- = ff{lu 4- wv -f nw) pdS (1) 

* The ezprefisions (8) and (9) were obtained by elementary reasoning by J. J. Thomson, 
On the Motion of Vortex Rings (Adams Prize Essay), London, 1883, pp. 5, 6, and the formulae (6) 
deduced from them, with the opposite signs, however, in the case of L, M, N, The correction 
is due to Mr Welsh. 

An interesting test of the formnlae as they now stand i» afforded by the case of a spherical mass 
rotating as if solid and surrounded by fluid at rest, provided we take into account the spherical 
vortex-sheet which represents the discontinuity of velocity. 

L. H. 14 



210 Vortex Motion [chap, vn 

If the surface S enclose all the vortices, we may put 

E.|_l^H.,(0 ,2, 

and it easily follows from Art. 150 (4) that at a great distance R from the 
vortices p will be finite, and lu -{- mv -{- nw of the order fi-', whilst when the 
surfacQ S is taken wholly at infinity, the elements 8S vary as iZ*. Hence, 
ultimately, the right-hand side of (1) vanishes, and we have 

r = const (3) 

We proceed to investigate one or two important kinematical expressions 
for T, still confining ourselves, for simplicity, to the case where the fluid 
(supposed incompressible) extends to infinity, and is at rest there, all the 
vortices being within a finite distance of the origin. 

The first of these expressions is indicated by the electro-magnetic analogy 
pointed out in Art. 148. Since 6 = 0, and therefore <E> — 0, we have 

22* = /) JJJ(m« + v* + w*) dxdydz 

by Art. 148 (4). The last member may be replaced by the sum of a surface- 
integral 

p ll{F (mw — nv) -\- [nu — Jw?) 4- i? (fc — rnu)} dS, 

and a volume-integral 

At points of the infinitely distant boundary, F, 0, H are ultimately of the 

order R-\ and w, v, w of the order fi-*, so that the surface-integral vanishes, 

and we have 

T = ipJ/K^I + Gr, + Hi;) dxdydz, (4) 

or, substituting the values of F, 0, H bom. Art. 148 (7), 

^ " iirlJIIJI^^' ^ ^r^ ^ - dxdydzdx'du'dz', (5) 

where each volume-integration extends over the whole space occupied by 
the vortices. 

A slightly different form may be given to this expression as follows. 

Regarding the vortex-system as made up of filaments, let Ss, Ss' be elements 

of length of any two filaments, a, a the corresponding cross-sections, and 

w, iii the corresponding vorticities. The elements of volume may be taken j 

to be ahs and a'S^', respectively, so that the expression following the integral 

signs in (5) is equivalent to 

cos € 5 , ,5 , 

. ii}Q08 , <JJ a OS , 



153 j Energy of a Vortex-System 211 

where c is the angle between 8* and S«'. If we put cja == #c, coV = #c', we 
have 



T^;l^JlKK'jj^ dads', (6) 



where the double integral is to be taken along the axes of the filaments, 
and the summation S includes (once only) every pair of filaments which 
are present. 

The faotor of p in (6) is identioal with the expression for the energy of a system of 
electric currents flowing along oonduotors coincident in position with the vortex-filaments, 
with strengths k, k, ... respectively*. The above investigation is in fact merely an 
inversion of the argument given in treatises on Electro-magnetism, whereby it is proved 
that 

^ 2 it'll ^ dads' =i III (a* +/8« -h /) dxdydz, 

4, i' denoting the strengths of the currents in the linear conductors whose elements are 
denoted by bs, ha', and a, /3, y the components of magnetic force at any point of the field. 

The theorem of this Art. is purely kinematical, and rests solely on the assumption that 
the functions u, v, w satisfy the equation of continuity, 

dx dy dz" ' 

throughout infinite space, and vanish at infinity. It can therefore by an easy generaliza- 
tion be extended to a case considered in Art. 144, where a liquid is supposed to circulate 
irrotationally through apertures in fixed solids, the values of u, v, w being now taken to be 
zero at all points of space not occupied by the fluid. The investigation of Art 151 shews 
that the distribution of velocity thus obtained may be regarded as due to a system of 
vortex-sheets coincident with the bounding surfaces. The energy of this system will be 
given by an obvious adaptation of the formula (6) above, and will therefore be proportional 
to that of the corresponding system of electric current-sheets. This proves a statement 
made by anticipation in Art. 144. 

Under the circumstances stated at the beginning of Art. 152, we have 
another usqjhil expression for T; viz. 

^ T = piSi{u{yC-zri) + v{zi-xO-{-w(xri-yi)}dxdydz1[. ...(7) 

To verify this, we take the right-hand member, and transform it by the 
process already so often employed, omitting the surface-integrals for the same 
reason as in the preceding Art. The first of the three terms gives 

'' ///" HI - 1) - Ki^ - ©} ^""^y^' 

= — pl\l\(vy -{-tjoz)-^ w* ■ dxdydz. 

* The 'rational' system of electrical units being understood; see ante p. 203. 
t Motion of Fluids, Art. 136 (1879). 

14—2 



212 Vortex Motion [chap, vn 

Trausf orming the remaining terms in the same way, adding, and making use 
of the equation of continuity, we obtain 

or, finally, on again transforming the last three terms, 

ip SSR'^^ + v^ + w^) dxdydz. 

In the case of a finite region the surface-integrals must be retained*. 
This involves the addition to the right-hand side of (7) of the term 

P JJ{(^ -h wv -h nw) {xu + yv -{- zw) — J (Za; -f my -h nz) g*} dS, ... (8) 
where g* = w* -h v* -h wK This simplifies in the case of affixed boundary. 

The value of the expression (7) must be unaltered by any displacement 
of the origin of co-ordinates. Hence we must have 

/JJ(vJ — ^) dxdydz = 0, i!i(wi — u^) dxdydz = 0, //J(wiy — t?^) dxdydz = 0, 

(9) 

These equations, which may easily be verified by partial integration, foUow also from 
the consideration that if there are no extraneous forces the components of the impulse 
parallel to the co-ordinate axes mu9t be constant Thus, taking first the case of a fluid 
enclosed in a fixed envelope of finite size, we have, in the notation of Art. 152, 

P = pjjjudxdydz -p J/^<Mf, (10) 

if ^ denote the velocity-potential near the envelope, where the motion is irrotationaL 

Hence ^=p j j j^dxdydz-p j jr^dS 

= -P j j j^^dxdydz+p j j j (v(-wn)dxdydz-pjjl^dS, (11) 

by Art. 146 (5). The first and third terms of this cancel, since at the envelope we have 
X =^l^f ^y '^'t. 20 (4) and Art. 146 (6). Hence for any re-entrant system of vortioea 
enclosed in a fixed vessel, we have 

-^ =P Hi W -WTiy) dxdydz, (12) 

with two similar equations. It ha.s been proved in Art. 119 that if the containing vessel 
be infinitely large, and infinitely distant from the vortices, P is constant. This gives the 
first of equations (9). 

Conversely from (9), established otherwise, we could infer the constancy of the com- 
ponents P, Q, R of the impulse*. 

* J. J. Thomson, he. ante p. 209. 



153-154J Two-Dimensional Theory 213 

RectiJinea/r Vortices. 

154. When the motion is in two dimensions cr, y we have w ^Q, whilst 
t*, V are functions of cr, y, only. Hence f = 0, ly = 0, so that the vortex-lines 
are straight lines parallel to z. The theory then takes a very simple form. 

The formulae (8) of Art. 148 are now replaced by 

~" dx dy' ~ dy dx' ' 

the functions <f>, tff being subject to the equations 

V,V-~fl, V,V-f, (2) 

and to the proper boundary-conditions. 

In the case of an incompressible fluid, to which we wiU now confine our- 
selves, we have 

»-i- «-i. ") 

where ^ is the stream-function of Art. 59. It is known from the theory 
of Attractions that the solution of 

ViV=^. (4) 

i being a given function of Xy y, is 

1^ = ^//riogr<fe'dy' + i^o. (5) 

where £' denotes the value of ^ at the point {x\ y% and r stands for 

{(X - x^)* + (y- y')«}*. 
The 'complementary function' ^o '^^y ^® ^^7 solution of 

Vl^o = 0; (6) 

it enables us to satisfy the boundary-conditions. 

In the case of an unlimited mass of Uquid, at rest at infinity, ^o ^ ^^^' 
stant. The formulae (3) and (5) then give 

Hence a vortex-filament whose co-ordinates are x', y' and whose strength is k 
contribates to the motion at (z, y) a velocity whose components are 

_ K_ y-y; , K_ x-x' 



214 Vortex Motion [chap, vn 

This velocity is perpendicular to the line joining the points (a, y), (x\ y% 
and its amount is /c/277r. 

Let us calculate the integrals iiuldxdy, and Hv^dxdyy where the integra- 
tions include all portions of the plane ay for which f does not vanish. We 
have 

Iju^dxdy = - ^jjjja'y^ dxdydx'dy', 

where each double integration includes the sections of all the vortices. Now, 
corresponding to any term 

a'^-^dxdycbfdy' 

of this result, we have another 

^'l^dxdydx'dy\ 

and these two neutralize each other. Hence, and by similar reasoning, 

jiu^dxdy = 0, Sfv^dxdy = (8) 

If as before we denote the strength of a vortex by #c, these results may 
be written 

S/m = 0, S#rt; = (9) 

Since the strength of each vortex is constant with regard to the time, the 
equations (9) express that the point whose co-ordinates are 

^^-=27' y-S (^^) 

is fixed throughout the motion. 

This point, which coincides with the centre of inertia of a film of matter 
distributed over the plane xy with the surface-density J, may be called the 
'centre' of the system of vortices, and the straight line parallel to z of which 
it is the projection may be called the 'axis' of the system. If Zic = 0, the 
centre is at infinity, or else indeterminate. 

155. Some interesting examples are furnished by the case of one or 
more isolated vortices of infinitely small section. Thus : 

1°. Let us suppose that we have only one vortex-filament present, and 
that the vorticity ^ has the same sign throughout its infinitely small 
section. Its centre, as just defined, will lie either within the substance of 
the filament, or infinitely close to it. Since this centre remains at rest, the 
filament as a whole will be stationary, though its parts may experience 
relative motions, and its centre will not necessarily lie always in the same 
element of fluid. Any particle at a finite distance r from the centre of the 
filament will describe a circle about the latter as axis, with constant velocity 






154-156] 



Vortex-Pair 



215 



KJimT, The region external to the vortex is doubly-connected; and the 
circulation in any (simple) circuit embracing it is of course k. The 
irrotational motion of the surrounding fluid is the same as in Art. 27 (2)« 

2°. Next suppose that we have two vortices, of strengths /Ci, /Cj, respec- 
tively. Let A^ B be their centres, the centre of the system. The motion 
of each filament as a whole is entirely due to the other, and is therefore 
always perpendicular to AB. Hence the two filaments remain always at the 
same distance from one another, and rotate with constant angular velocity 
about 0, which is fixed. This angular velocity is easily found; we have 
only to divide the velocity of A (say), viz. /C2/(27r . AB), by the distance -40, 
where 



A0 = 



Ki 4- /Cj 



AB, 



and so obtain 



Ki + K^ 

2n.AB^' 



If #ci, K^ be of the same sign, i.e. if the directions of rotation in the two 
vortices be the same, lies between A and B; but if the rotations be of 
opposite signs, Ues in AB, or BA, produced. 

If /Ci = — /Cj, is at infinity; but it is easily seen that A, B move with 
equal velocities /Ci/(27r . AB) at right angles to AB, which remains fixed in 
direction. Such a combination of two equal and opposite vortices may be 
called a 'vortex-pair.' It is the two-dimensional analogue of a circular 
vortex-ring (Art. 160), and exhibits many of the properties of the latter. 




The stream-lines of a vortex-pair form a system of coaxal circles, as shewn 
on p. 68, the vortices being at the limiting points {± a, 0). To find the 



216 Vortex Motion [chap, vn 

relative stream-lines, we superpose a general velocity equal and opposite to 
that of the vortices, and obtain, for the relative stream-function, 



-&te+>»*r;) m 



in the notation of Art. 64, 2^. The figure (which is turned through 90° for 
convenience) shews a few of the lines. The line ^ =: consists partly of the 
axis of y, and partly of an oval surrounding both vortices. 

It is plain that the particidar portion of fluid enclosed within this oval 
accompanies the vortex-pair in its career, the motion at external points 
being exactly that which would be produced by a rigid cylinder having 
the same boundary; cf. Art. 71. The semi-axes of the oval are 2*09 a and 
1*73 a, approximately*. 

A difficulty is sometimes felt, in this as in the analogous instance of a vortex-ring, 
in understanding why the vortices should not be stationary. If in the figure on p. 68 
the filaments were replaced by solid cylinders of small circular section, the latter might 
indeed remain at rest, provided they were rigidly connected by some contrivance which 
did not interfere with the motion of the fluid; but in the absence of such a connection 
they would in the first instance be attracted towards one another, on the principle 
explained in Art. 23. This attraction is however neutralized if we superpose a general 
velocity V of suitable amount in the direction opposite to the cyclic motion half-way 
between the cylinders. To find F, we remark that the fluid velocities at the two points 
(a ±^ c, 0), where c is small, will be approximately equal in absolute magnitude, provided 

V '\-— ^ =_!L +_!! V 

2irC 4)ra 2irC ^na ' 
where k is the circulation. Hence 



F = 



47ra' 



which is exactly the velocity of translation of the vortex-pair, in the original form of the 
problem t* 

Since the velocity of the fluid at all points of the plane of symmetry is 
wholly tangential, we may suppose this plane to form a rigid boundary of 
the fluid on either side of it, and so obtain the case of a single rectilinear 
vortex in the neighbourhood of a fixed plane wall to which it is parallel. 
The filament moves parallel to the plane with the velocity KJifirh, where h is 
the distance from the wall. 

Again, since the stream-lines are circles, we can also derive the solution 
of the case where we have a single vortex-filament in a space bounded, either 
internally or externally, by a fixed circular cyUnder. 

♦ Cf. Sir W. Thomaon, "On Vortex Atoms," PhU. Mag, (4), t. xxxiv. p. 20 (1867) [PaperB, 
t. iv. p. 1]; and Biecke, Q6U, Nachr. 1888, where paths of fluid particles are also delineated. 

t A more exact investigation is given by Hicks, "On the Condition of Steady Motion of Two 
Cylinders in a Fluid," Quart. Jaum, Maih. t. xvii. p. 194 (1881X 



155] 



Method of Images 



217 




Thus, in the figure, let EPD be the section of the cylinder, A the position of the vortex 
(supposed in this case external), and let B be the * image' of A with respect to the circle 
EPD, viz. C being the centre, let 

CB . CA =c«, — ^ 

where c is the radius of the circle. If P be any point on 
the circle, we have 

AP AE AD 
BP^m^BD^'^'^^''^ 

so that the circle occupies the position of a stream-line due 

to a vortex-pair at A, B, Since the motion of the vortex A would be perpendicular to AB, 
it is plain that all the conditions of the problem will be satisfied if we suppose A to 
describe a circle about the axis of the cylinder with the constant velocity 

K __ K . CA 

' 2w . AB" ^ 2w {CA^ 'C*y 

where k denotes the strength of A, 

In the same way a single vortex of strength k, situated inside a fixed circular cylinder, 
say at B, would describe a circle with constant velocity 

K,CB 

It is to be noticed, however*, that in the case of the external vortex the motion is not 
completely determinate unless, in addition to the strength k, the value of the circulation 
in a circuit embracing the cylinder (but not the vortex) is prescribed. In the above 
solution, this circulation is that due to the vortex-image at B and is - k. This may be 
annulled by the superposition of an additional vortex + ic at (7, in which case we have, for 
the velocity of A, 



,CA 



kc^ 



For a prescribed circulation k we must add to this the term ic72fr . CA, 

3^. If we have four parallel rectilinear vortices whose centres form a 
rectangle ABB'A\ the strengths being #c for the vortices A\ S, and — k for the 




vortices A^ B\ it is evident that the centres will always form a rectangle. 
Further, the various rotations having the directions indicated in the figure 

* F. A. Tarieton, "On a Problem in Vortex Motion," Proc B, I. A. December 12, 1892. 



218 Vortex Motion [chap, vn 

we see that the e£Eect of the presence of the pair A, A' onB, ff iBto separate 
them, and at the same time to diminish their velocity perpendicular to the 
line joining them. The planes which bisect AB, AA' at right angles may 
(either or both) be taken as fixed rigid boundaries. We thus get the case 
where a pair of vortices, of equal and opposite strengths, move towards (or 
from) a plane wall, or where a single vortex moves in the angle between two 
perpendicular waUs. 

If ;i:, ^ be the co-ordinates of the vortex A relative to the planes of symmetry, we 
readily find 

• - * ^* • - * y* /o\ 

^-"i^P' ^-Tn'^' ^^^ 

where r^=x^+y^. By division we obtain the differential equation of the path, viz. 

dx dy 

whence o* (a;* + y*) = 4a:* y*, 

a being an arbitrary constant, or, transforming to polar co-ordinates, 

r=-T^ (3) 

Also since o:y -yx^— , 

the vortex moves as if under a centre of force at the origin. This force is repulsive, and 
its law is that of the inverse cube*. 

156. If we write, as in Chapter iv., 

2 = 35 + iy, U7 = <^ 4- i^, (1) 

the potential- and stream-functions due to an infinite row of equidistant 
vortices, each of strength #c, whose co-ordinates are 

(0, 0), (± a, 0), (± 2a, 0), . . . , 

will be given by the formula 

t£; = 2;^logsm-; (2) 

cf. Art. 64, 3"*. This makes 

dw tK ^ rrz .«. 

w - tv = - -V- = - s- cot — , (3) 

dz 2a a 

• Greenhill, "On Plane Vortex-Motion," Quart. Joum. Math, t. xv. (1887); Grobli, Die 
Bewegung paraUeler geradliniger Wirhdfdden, Zurich, 1877. These papers contain other in- 
teresting examples of rectilinear vortex-Bystems. The case of a system of equal and parallel 
vortices whose intersections with the plane xy are the angular points of a regular polygon was 
treated by J. J. Thomson in his Motion of Vortex Rings, pp. $f4.... He finds that the 
configuration is stable if, and only if, the number of vortices does not exceed six. For some 
further references as to special problems see Hicks, BriL Ass, Rep. 18)^2, pp. 41 ... ; Love, Ix, 
anJte p. 183. 

An ingenious method of transforming plane problems in vortex -motion was given by Routh» 
'*Some Applications of Conjugate Functions," Proc. Lond. Math. Soe. t. xi]> p. 73 (1881). 



155-166] 



Rows of Vortices 



219 



whence 



U^ — TT- 



sinh {^tryja) 



V^TT- 



sin (2mxla) 



2a cosh {2ny/a) — cob {2nrx/a) ' " 2a cosh {2ny/a) — cos (2w«/a) ' 

(4) 

These expressions make w = =f J /c/a, v = 0, f or y = ± oo ; the row of vortices 
is in fact, as regards distant points, equivalent to a vortex-sheet of uniform 
strength #c/a (Art. 161). 




The diagram shews the arrangement of the stream-lines. 

It follows easily that if there are two parallel rows of equidistant vortices, 
symmetrical with respect to the plane y = 0, the strengths being #c for the 
upper and — #c for the lower row, as indicated on the next page, the whole 
system will advance with a uniform velocity 

U^^coth-, (5) 

2a a ^ ' 

where b is the distance between the two rows. The mean velocity in the 
plane of symmetry is /c/a. The velocity at a distance outside the two rows 
tends to the Umit 0. 

If the arrangement be modified so that each vortex in one row is opposite 
the centre of the interval between two consecutive vortices in the other row, 
as shewn on p. 222, the general velocity of advance is 



2a a 



(6) 



The mean velocity in the medial plane is again x/a. 

The stability of these various arrangements has been discussed by von K&rm4n*. 
Taking first the case of the single row, let us suppose the vortex whose undisturbed 
co-ordinates are {ma, 0) to be displaced to the point {ma +Xm» ^m)* ^o formulae of 
Art. 154 give 

(7) 



^0 _ « « yp-ym 






where 



2ff m *'m 

rn? = (a^o - a:„, - ma)^ + {y^ - yJi\ 



(8) 



* "FIuBsigkeits- u. Luftwiderstand," Phya, Zeitachr, t. xiii (1912) ; also Om, Nachr. 1912, p. 547. 
The investigation is only given in outline in these papers; I have supplied various steps. 



220 Vortex Motion [chap, vn 

and the summation with respect to m inoludes all positive and negative integral values, 
zero being of ooorse excluded. If we neglect terms of the second order in the displacements, 
we find 

The first term in the value of dyo/dt is to be omitted as being independent of the 
disturbance*. 



Consider now a disturbance of the type 

x«=ae**~*, yn.=/3e**^, (10) 

where <f> may be assumed to lie between and 2ir. If ^ be small this has the character of 
a wave of length 27ra/^. We find 

|=-Xft f=-X„ (11) 

, X K /1-C08d> 1 -COS 20 l-cos3d> \ « , ,« .. ,,«v 

where X=— ,( ^ ^, ^ + ^T-^ + 3«-^ + ' * 'j =£?*(2ir -0). ..(12) 

The arrangement is therefore unstable, the disturbance ultimately increasing as e^^ When 
the wave-length is large comi)ared with a we have 

X =iK<f>la*, (13) 

approximately; cf. Art. 234. 

Proceeding next to the case of the symmetrical double row, the positions at time t of 
vortices in the upper and lower rows may be taken to be 

(wia + Ut + x^, \h + ym), and (wa + C7« + re/, - ^6 + y/)* 

respectively, where TJ denotes the general velocity of advance of the system, and the origin 
is in the plane of symmetry. 

<3) c?) <5) 6) 



(J (J) <j) (J 

The component velocities of a vortex in the upper row, e.g, that for which m = 0, due 
to the remaining vortices of the same row, will be given as before by (9), where the sum 
1m~^ may be omitted. The components due to the vortex n of the lower row will be 

2n r„« ' " 27r f^* 

where r„« = (xo - Xn - na)^ + (yo - y«' + b)K 

If we n^lect terms of the second order in the disturbance we find, after a little reduction, 

2iuib 

^!(^?5^TT»T«^'^^-'^'*'> ^^^^ 

K (ft - " « m2a« *** n (w*a« + 6«)« ^^^^ " ^" ^ 

* In the summations the vortices are to be taken in pairs equidistant from the origin ; otherwise 
the result would be indeterminate. The investigation may be regarded as applying to the central 
portions of a long, bat not infinitely long, row; the term referred to is then negligible. 



156] Stability of Double Mows 221 

where the summations with respect to n go from - ao to + oo , including zero. The terms 
in (14) independent of the disturbance will cancel, since, by (5) 

C7= 5-COth — = gr-2 a , . ., . 

If we now put 

ar,^=ae'«*, ym = /3e'«*, ar/ = a'e"^, y/ = i8 V^ (16) 

where < ^ < 2ir, the equations take the form 

?!«!^=_^„ -Ca' + ^/s-.l 

If we write, for shortness, 

k = b/a, (18) 

the values of the coefficients are* 

^ = 1^?-- !(-^?Ti.). = 4*<2.-*) + 55g^ (19) 

_ ^ 2nke*^ _ fir<^ cosh fc (w - <^) g« sinh l;<f) l 

»(n« + ifc*)«" 1 sinhifcir " sinh'ijir J ' ^^"^ 

(n« - l^) e*"^ _ ir» cosh k<l> ff<^ sinh ifc (w - <^) 
^^n (»« + ik*)« """ sinh«ifcir " sinhifeir ^^^^ 

To deduce the equations relating to the lower row we have merely to reverse the signs 
of K and 6, and to interchange accented and unaccented letters. Hence 

" ^' \ (22) 

^ = -4a' + Ca + Bfi. 

K at J 

The formulae (17) and (22) are the equations of motion of the vortex-system in what may 
be called a normal mode of the disturbance. 

The solutions are of two types. In the first type we have 

a^a\ i8= -i8' (23) 

and therefore 

* I (24) 

The solution involves exponentials e^^ the values of X being given by 

^^X= 'B±J(A^-C^) .' (26) 

K 

* The summations with respect to n can be derived from the Fourier expansion 

oo8hifc(T~0) _ 1 jl 21; COB » 21; COB 2^ 1 

Bmhl;ir "«■ t* 1*+** "*" 2»+i* +•••[• 



222 Vortex Motion [chap, vn 

In the seoond type we have 

a^-a', /3=/3' (26) 

and therefore -^ jt = Ba - (A + C)ff, ] 

' * I (27) 

___=_(^.C)a+mJ 

The corresponding values of X are given by 

?^ X = B±^(A* - C«) (28) 

Since B is a pure imaginary, whilst A and C are real, it is necessary for stability in 

each case that A* should not exceed C for admissible values of (f>. Now when ^ = n- we 

find 

u4 + C = Jw* tanh« Jifcir, A - C = in* coth« ikir, (29) 

so that A* - C^iB positive. We conclude that both types are unstable. 

Passing to the unsymmetrical case, we denote the positions of the displaced vortices by 

{ma + F« + Xn,f ib + y^), and ((n + i)a -\- Vt + Zn, - J6 + yn), 

where F is given by (6). The requisite formulae are obtained by writing n + ^ f or n in 
preceding results. 

<J <3) <3) <5) 



<3) ^ ^ 

The equations (17) and (22) will accordingly apply, provided* 

1 - e**^ ^ (w + i)* - it« - . ,- ^, «•« 

{2n+ 1) fce*(»+*)^ _ . f 7r<^8inhA;(7r-<f>) n-^sinh^l 
» {(n + f)« + ^»}« ""*( cosh^TT "*" coshH-irJ' ^^^^ 

p , ((w + i)' - ^} e^ ^"""^^ * _ ir« cosh A;<» ir<<) cosh fe (,r - ») 

» {(w + i)* + ^}* "" oo8h« kn ' cosh ifctr ^"^^^ 

These values of ^, JS, C are to be substituted in (25) and (28). As in the former case it is 
necessary for stability that A* should not be greater than C*. Now when ^ = ir, C = 0; 
hence A must also vanish, or 

C08h«ifcir = 2, ifcn- = -8814, 6/o=ifc = -281 (33) 

The configuration is therefore unstable unless the ratio of the interval between the two rows 
to the distance between consecutive vortices has precisely this value. 

To determine whether the arrangement is stable, under the above condition, for all 
values of ^ from to 2Yr, let us write for a moment k(ir - <t)) = x, kv = fi, bo that 

J^A = - ioc*, k^C = i(fix cosh fix cosh a; - /n* sinh fj, sinh x) ( 34) 



* The summations with respect to n can be derived from the ezpansion 

sinh A; (t - 0) _ 2 \k cos i<p , k cos |^ 
cosh ibr 



2 \ k cos i<p k cos 1^ , j 
"i^ l(i)* + it«+(i)rnbS + ---r 



166-157] Stability of Double Mows 223 

where x may range between ±fi. Since ^ is an even and C an odd function of a;, it is 
sufficient for comparison of absolute values to suppose x positive. Hence, writing 

y=fi cosh fi cosh z - fi^ sinh /n x, .(36) 

X 

we have to ascertain whether this is positive for <x </i. Since /li = •8814, cosh /x = ^2, 
sinh fi= If yjs positive for x = 0, and it evidently vanishes for x= /a. Again 

dy , • u . « • u sinh a; , . , coshx , ,«^, 

^ = /i cosh fi smh « + /i* smh fi — j /i* smh/x 1, (36) 

which is equal to - 1 f or x = 0, and vanishes toTX = fi. Finally, 

tPy , , • • u sinh a; ^ - . , cosh a? « • . , sinh a; ,„_. 

;t4 = /* cosh /i cosh X - /Li* sinh ;i- — + 2/Li* sinh /i — -^-z 2/i' smh /a , , . .(37) 

(M/ a? X a» 

which is easily seen to be positive for all values of x, since (tanha;)/a;<l. Hence as x 
increases from to /^ dy/dx is steadily increasing from - 1 to 0, and is therefore n^ative. 
Hence y steadily diminishes from its initial positive value to zero, and is therefore positive. 

We conclude that the configuration is definitely stable* except for a: = ± /i, when (^ = 
or 2ir, in which cases B = 0, by (31), and therefore X = 0. Since the disturbed particles 
are then all in the same phase, the reason why the period of disturbance should be 
infinite is easily perceived. 

■ 

157. When, as in the case of a vortex-pair, or a system of vortex-pairs, 
the algebraic sum of the strengths of all the vortices is zero, we may work 
out a theory of the impulse,' in two dimensions, analogous to that given in 
Arts. 119, 152 for the case of a finite vortex-system. The detailed exami- 
nation of this must be left to the reader. If P, Q denote the components of 
the impulse parallel to x and y, and N its moment about Oz, all reckoned per 
unit depth of the fluid parallel to z, it will be found that 

P = pny^dxdy, Q = -pSSxCdxdyA . 

N^--\pll{x^ + y^)ldxdy. J 

For instance, in the case of a single vortex-pair, the strengths of the two 
vortices being ± k, and their distance apart c, the impulse is pKC, in a line 
bisecting c at right angles. 



j (2) 



The constancy of the impulse gives 

^KX = const., J^Ky = const., 
S/c (x^ + y^) = const. 

It may also be shewn that the energy of the motion in the present case 
is given by 

T = ^yss^^dxdy=-y:s:K^ (3) 

When Sic is not zero, the energy and the moment of the impulse are both 
infinite, as may be easily verified in the case of a single rectilinear vortex. 

* This is stated without proof by KdrmiLn. 



(4) 



224 Vortex Motion [chap, vn 

The theory of a system of isolated rectilinear vortices has been put in a very elegant 
form by Kirchhoff*. 

Denoting the positions of the centres of the respective vortices by (z^, yi)f (x^» y^iy . . . 
and their strengths by k^, k,, . . . , it is evident from Art. 154 that we may write 

da^_ W dy^ dW 

"^dt^'dy^* ""(ft'ax, ' 

where W =^ 2*, «, log r„, .., (6) 

if fit denote the distance between the vortices k^, k^. 

Since W depends only on the rdaiive configuration of the vortices, its value is unaltered 
when Xi, x^, ... are increased by the same amount, whence id W/dxi =0, and, in the same 
way, 2dW/dyi=0, This gives the first two of equations (2), but the proof is not now 
limited to the case of 2k =0. The argument is in fact substantially the same as in 
Art. 154. Again, we obtain from (4) 



^ / dx dy\ ^/ dW dW\ 



or if we introduce polar co-ordinates (r^, ^i), (r,, B^), ... for the several vortices, 

^"'S^'^^ (^> 

Since W is unaltered by a rotation of the axes of co-ordinates in their own plane about the 
origin, we have tdW/dO =0, whence 

2«cr*=const., (7) 

which agrees with the third of equations (2), but is free from the restriction there implied. 
An additional integral of (4) is obtained as follows. We have 



^ f dy dx\ ^/ dW dW\ 



or ^""^dt^^^'W (®) 

If every r be increased in the ratio 1 +c, where c is infinitesimal, the increment of IF is 
equal to Scr . dW/dr, But since the new configuration of the vortex-system is geometrically 
similar to the former one, the mutual distances r^ are altered in the same ratio 1 +f, and 
therefore, from (5), the increment of IF is f/2n- . Ik^k^, Hence (8) may be written in the 
form 

^'''^di^2ir^''^''' (^) 

158. The preceding results are independent of the form of the sections 
of the vortices, so long as the dimensions of these sections are small compared 
with the mutual distances of the vortices themselves. The simplest case 
is of course when the sections are circular, and it is of interest to inquire 
whether this form is stable. This question has been examined by Kelvin f. 

* Jiechanik, c. zx. 

t Sir W. Thomson, "On the Vibrations of a Columnar Vortex," Phil Mag. (6), t. x. p. 156 
(1880) [Papers, t. iv. p. 152]. 



167-158] Stability of a Cylindrical Vortex 225 

WEen'the distorbaiioe is in two dimensions only, the calculations are veiy Simple.. Let 
us suppose, as in Art. 27, that the space within a circle r—a, having the centre as origin, 
is occupied by fluid having a uniform vorticity o>, and that this is surrounded by fluid 
moving irrotationally. If the motion be continuous at this circle we iiave, for r <.ay 

^=-J«(a*-r«) (1) 

while for r > a, ^ = - Jwa* log a/r. (2) 

To examine the effect of a slight irrotational disturbance, we assume, for r K^a, 

■^ = - J« (o* -r*) +A — 008 (t6 - <r<).] 

" I (3) 

and, for r> a, ^= -^a*log -+-4 — cos(«^-aO>j 

where 9 is integral, and o- is to be determined. The constant A must have the same 
value in these two expressions, since the radial component of the velocity, -d^/rd^, must 
be continuous at the boundary of the vortex, for which r^a^ approximately. Assuming 
for the equation to this boundary 

r =a +a cos («^ -crO. (4) 

we have still to express that the transverse component (d^/Br) of the velocity is continuous. 
This gives 

Substituting from (4), and neglecting the square of a, we find 

wi^-ZaA/a (5) 

So far the work is purely kinematical; the dynamical theorem that the vortex-lines 
move with the fluid shews that the normal velocity of a particle on the boundary must be 
equal to that of the boundary itself. This condition gives 



8r_ 9^ dyjt dr 



where r has the value (4), or 



A . 8a ,^. 

<ra=« — +4<»a. — (o) 



Eliminating the ratio A/a between (5) and (6) we find 

<7=J(«-l)a) (7) 

Hence the disturbance represented by the plane harmonics in (3) consists of a system 
of corrugations travelling round the circumference of the vortex with an angular velocity 

<r/«= (« -!)/«. Jft) (8) 

This is the angular velocity in space; relative to the rotating fluid the angular 
velocity is 

v/8-^= -i»l8, (9) 

the direction being opposite to that of the rotation. 

When 9=2, the disturbed section is an ellipse which rotates about its centre with 
angular velocity ^«». 

The transverse and longitudinal oscillations of an isolated rectilinear vortex-filament 
have also been discussed by Kelvin in the paper cited. 

I1.H. 16 



226 Vortex Motion [chap, vn 

159. The particular caae of an elliptic disturbance can be solved without 
approximation as follows*. 

Let us suppose that the space within the ellipse 

?!+y"=i (1) 

is occupied by liquid having a uniform vorticity a>, whilst the surrounding fluid is moving 
irrotationally. It will appear that the conditions of the problem can all be satisfied if we 
imagine the elliptic boundary to rotate, without change of shape, with a constant angular 
velocity (n, say), to be determined. 

The formula for the external space can be at once written down from Art. 72, 4^ ; viz. 
we have 

^ = Jw (a +6)« t-^ cos 2»7 + JcDO^jf (2) 

where (, 17 now denote the elliptic co-ordinates of Art. 71, 3°, and the cyclic constant k has 
been put =9ra6a). 

The value of ^ for the internal space has to satisfy 






3a:* 3y^ 
with the boundary-condition -7 + rj = - wy . -^ +na; . ^ (4) 

These conditions are both fulfilled by 

^ = J« (At^ ^By^) (6) 

provided A +JB = 1, Aa^ -JB6«=-(a« -6*) (6) 

It remains to express that there is no tangential slipping at the boundary of the 
vortex; ».e. that the values of 3^/8$ obtained from (2) and (5) there coincide. Putting 
x=c cosh f cos f)ty=c sinh f sin 17, where c — J(a*- 6*), differentiating, and equating coeffi- 
cients of cos 2i7, we obtain the additional condition 

-in (a +6)* e"*^ =\(ac^ (A -B) cosh J sinh f, 

where ( is the parameter of the ellipse (1). This is equivalent to 

.^-B=-».?!^* (7) 

since, at points of the ellipse, cosh ( =alCt sinh ( = 6/c. 

Combined with (6) this gives . Aa -Bh = — 7, (8) 

•"■^ "=(^«» <®> 

When a =6, this agrees with our former approximate result. 

The component velocities x, y oi a particle of the vortex relative to the principal axes 
of the ellipse are given by 

x= - ^ +ny, y=^'-nz, 
cy " '^ ex 

* KirchhoiT, Mechanik, c. xz. p. 261; Basset, Hydrodynamics, t. ii. p. 41. 



»• . 



169.-161] Elliptic Vortex 227 

j» V V X 

whence we find -=-n|, f = »-^ (10) 

a o a 

Int^^ting, we find a; =fai cos (rrf + c)^ y==tt8in(n<+«), (11) 

where A;, c are arbitrary constants, so that the relative paths of the particles are ellipses 
similar to the boundary of the vortex, described according to the harmonic law. If x', f/ 
be the co-oidinates relative to axes fixed in space, we find 

x' =x oos fU-ymnnt=ik{a+b) cos (2nt + c) + ^ib (a - b) cos c. 



+ f)+ih{a-b)<iOBtA 
+ <) -ik{a-b) sine./ 
The absolute paths are therefore circles described with angular velocity 2n*. 



(12) 
i/^xwBtU+y cos tU =^k (a +6) sin {2rU 



160. It was pointed out in Art. 80 that the motion of an incompressible 
fluid in a curved stratum of small and imif orm thickness is completely defined 
by a stream-function iff, so that any kinematical problem of this kind may be 
transformed by projection into one relating to a plane stratum. If, further, 
the projection be *orthomorphic,' the kinetic energy of corresponding portions 
of liquid, and the circulations in corresponding circuits, are the same in the 
two motions. The latter statement shews that vortices transform into vor- 
tices of equal strengths. It follows at once from Art. 145 that in the case of 
a closed simply-connected surface the algebraic sum of the strengths of all 
the vortices present is zero. 

Let us apply this to motion in a spherical stratum. The simplest case is 
that of a pair of isolated vortices situate at antipodal points ; the stream-Unes 
are then parallel small circles, the velocity varying inversely as the radius 
of the circle. For a vortex-pair situate at any two points A, B, the stream- 
lines are coaxal circles as in Art. 80. It is easily found by the method of 
stereographic projection that the velocity at any point P is the resultant of 
two velocities ic/27ra . cot ^di and ic/27ra . cot |02> perpendicular respectively 
to the great-circle arcs AP^ BP, where d^, d^ denote the lengths of these arcs, 
a the radius of the sphere, and ± k the strengths of the vortices. The centre f 
(see Art. 164) of either vortex moves perpendicular to AB with a velocity 
Kl2jra . cot ^AB, The two vortices therefore describe parallel and equal small 
circles, remaining at a constant distance from each other. 

Circular Vortices, 

161. Let us next take the case where all the vortices present in the 
liquid (supposed unlimited as before) are circular, having the axis of 2; as a 
common axis. Let m denote the distance of any point P from this axis, v the 

* For further researches in this connection see Hill, **0n the Motion of Fluid part of which 
is moving rotationally and part irrotationally/' PkiL Trans. 1884; Love, "On the Stability of 
certain Vortex Motions," Proc Lond. Math, 80c, t. zxv. p. 18 (1893). 

t To prevent possible misconception it may be remarked that the centres of corresponding 
Tortices are not necessarily corresponding points. The paths of these centres are therefore not 
in general projective. 

16—2 



228 Vortex Motion [chap, vti 

velocity in the direction of w, and co the resultant vorticity at P. It is 
evident that u, v, oi are functions of Xy w only. 

Under these circumstances there exists a stream-function iff, defined as in 
Art. 94, viz. we have 

, ^??^_^=i f^V _L.^^].^\ . (2) 

" dx dm m \dx^ dm^ w dm) 

It is easily seen from the expressions (7) of Art. 148 that the vector 
{Fy Oy H) will under the present conditions be everjrwhere perpendicular to 
the axis of x and the radius m. If we denote its magnitude by S, the flux 
through the circle {Xy w) will be 27TmSy whence 

ilf=-mS (3) 

To find the value of ^ at (x, m) due to a single vortex-filament of cir- 
culation Ky whose co-ordinates are x\ to', we note that that element which 
makes an angle d with the direction of S may be denoted by w'S6, and there- 
fore by Art. 149 (1) 

. « Ktmn' [^ COB d jj^ ,.. 

^ = "^^=-1^Jo — ^' (*) 

where r = {(« - a?')* + ©« + to'* - 2ojot' cos ^*. (5) 

If we denote by fj, r, the least and greatest distances, respectively, of the 
point P from the vortex, viz. 

f i« = (a - xy + (id ~ ©')^ r,* = (» - xy + (m + to')*, ... (6) 
we have r « = r i^ cos* Jfl + r,* sin« J0, 4toro' cos 6^ r^ -f- r ,« - 2r«, . . (7) 
and therefore 

^ Stt L^*"^ "^ * ^ j V(ri« co8« ^e + V sii* P) 

- 2 / ' V(ri« cos« ^e + r,« sin« ^0)4$] (8) 

-'0 J 

The integrals are of the types met with in the theory of the 'arithmetico- 
geometrical mean.'* In the ordinary, less symmetrical, notation of 'com- 
plete' elliptic integrals we have 

*--£(«»■)'{(?-*)*•,(*)-?«,(»)} (9) 

prodded i. . , _ i; . j__^*^__,^. „o) 

The value of ^ at any assigned point can therefore be computed with the 
help of Legendre's tables. 

* See Gaylev, EUipitc Functions, Cambridge, 1876, c. ziii. 




161] Circular' Vortices 229 

A neater expression majt be obtained' by means of fLanden's trans- 
formation'*; thus 

^«-£(^i+T.){Ji(A)~£i(A)}, (11) 

provided A = ^— * (12) 

^j + ^1 

To verify this, let AB be a straight line divided at P into two segments PA, PB of 
lengths 7*1 , r,, respectively ; and describe the circle on AB as diameter. C being the centre, 
and Q any point on the circumference, let the angles QCA, QPA be denoted by B, S, respec- 
tively ; and draw CN perpendicular to QP. If PQ=r, we have 

r»=ri«co82i^+r,«sin*i^, rdS=CQM , cos CQN=QNM (13) 

Also CP.'^^^-oosCQN^^^fS. 

r r ^ r CQ 

and therefore 

^ co8^_ 6S PQds ^^ ds Qms pms f \q 

Hence I \/^^ 

since j PNdS^CP JQO3SdS=0. 

Now QN = ^{CQ^ - CP» sin« 5) = J (fi +rj) V(l - X' sin« 5), (16) 

« ^=?C=rT^; .........(16) 

The formula (14) may therefore be written 

i(r.-r,)f'?^^d^=2{^i(X)-^i(X)} (17) 

Jo r 

which brings (4) into the required form (11). 

The forms of the stream-lines corresponding to equidistant values of ^ are 
shewn on the next page. They are traced by a method devised by Maxwell, 
to whom the formula (11) is also duef. 

Expressions for the velocity-potential and the stream-function can also be 
obtained in the form of definite integrals involving Bessel's Fimctions. 

Thus, supposing the vortex to occupy the position of the circle ic » 0, 
to = a, it is evident that the portions of the positive side of the plane x^O 
which lie within and without this circle constitute two distinct equipotential 
surfaces. Hence, assuming that we have ^ = ^/c for x » 0, to < a, and ^ = 
for a; » 0, to > a, we obtain from Art. 102 (2) 

<t>^\Ka\ e-^'Jo{kw)J^{ka)dk, (18) 

J 

* See Cayley, 2.c. 

t EledricUy and Jiagneiism, Arto. 704, 705. See also Minchin, PhiL Mag. (6), t. zxx7. 
(1803); Nagaoka, PhO. Mag. (6), t. vi (1903). 



230 Vortex Motion 

and t^eiefoce, in accordance with Art. 100 (5), 

= - i Kom I e-*'Ji {km) J, (ka) dk. 



These formulae relate of course to the region x > 0*. 



[chap, vn 

(19) 




> The [onnnlk for ^ 




See alao Nkgaoka, Le. 



161-162] StreamrlAnes of a Vortex-Ring 231 

It was shewn in Art. 150 that the Value of ^ is that due to a system of 
double sources distributed with uniform density k over the interior of the 
circle. The values of <f> and tft for a uniform distribution of simple sources 
over the same area have been given in Art. 102 (11). The above formulae 
(18) and (19) can thence be derived by differentiating with respect to x, and 
adjusting the constant factor*. 

162: The energy of any system of circular vortices having the axis of x 
as a common axis, is 

= — Trp / j^Kodxdw = — npYiKift, . • (1) 

by a partial integration, the integrated terms vanishing at the limits. We 
have here used ic to denote the strength coSxStn of an elementary vortex- 
filament. 

Again the formula (7) of Art. 153 becomes f 

T =*= iirp //(row — xv) XDcodxdy = 2mpl!iKXD {mu — xv) (2) 

The impulse of the system obviously reduces to a force along Ox. 
By Art. 152 (6), 

-P ~ ip lliyt — ^) dxdydz = irp ijwhadxdm *= TrpSicw* (3) 

If we introduce the symbols Xq, Wq defined by the equations 

.these determine a circle whose position evidently depends on the strengths 
and the configuration of the vortices, and not on the position of the origin on 
the axis of symmetry. It may be called the 'circular axis' of the whole 
system of vortex-rings. 

Since k is constant for each vortex, the constancy of the impulse shews, 
by (3) and (4), that the circular axis remains constant in radius. To find its 
motion parallel to x, we have, from (4), 

Sic . ©0* . -j^ = S/ctD* ^ + 2Eicrox -^ = Ektd {mu 4- 2xu) (5) 

* Other ezpreadons for and ^ can he obtained in terms of zonal spherical harmonics. 
ThoB the value of is given in Thomson and Tait, Art. 546; and that of ^ can be deduced by 
the formulae (11), (12) of Art. 95 ante. The elliptic -integral forms are however the most useful 
for purposes of interpretation. 

t At any point in the plane z=0 we have y=iff, (=0, 17=0, i;=l<a,v==v; the rest follows by 
symmetry* 



232 Vortex Motion [chap, vn 

With the help of (2) this can be ptit in the fonn 

Sic. Too* . ^® = ^ + 32ic (a; -Zo)mv,.,. (6) 

where the added term vanishes, since JIkwv == on account of the constancy 
of the mean radius (mo). 

163. Let us now consider, in particular, the case of an isolated vortex- 
ring the dimensions of whose cross-section are small compared with th^ 
radius (tOo)* It has been shewn that 

* - - nil'- Citt) - *. CTrf;)} '" + '.) '»''^*^. ■ • m 

where r^, rj are defined by Art. 161 (6). For points («, in) in or near the 
substance of the vortex, the ratio r i/fs is small, and the modulus (A) of the 
elliptic integrals is accordingly nearly equal to. unity. We then have 

yi(A) = iIog^„ ^i(A)=l (2) 

approximately*, where A' denotes the complementary modulus, viz. 

A'2 = 1 - A2 = j-^^^^ ,- , (3) 

or A'* = ^Ti/fj, nearly. 

Hence at points within the substance of the vortex the value of ^ is of 
the order ict^o log (tOo/c), where e is a small linear magnitude comparable with 
the dimensions of the section. The velocities at such points, depending 
(Art. 94) on the differential coefficients of ^, will be of the order ic/c. 

We can now estimate the magnitude of the velocity dxQ/dt of translation 
of the vortex-ring. By Art. 162 (1), T is of the order pk^Dq log (otq/c), and v is, 
as we have seen, of the order k/c ; whilst x — a;© is of course of the order €. 
Hence the second term on the right-hand side of the formula (6) of the 
preceding Art. is, in the present case, small compared with the first, and the 
velocity of translation of the ring is of the order k/wq . log (mo/^)> ^^^ 
approximately constant. 

An isolated vortex-ring moves then, without sensible change of size, 
parallel to its rectilinear axis with nearly constant velocity. This velocity 
is smaU compared with that of the fluid in the immediate neighbourhood of 
the circular axis, but may be greater or less than ^k/wq, the velocity of the 
fluid at the centre of the ring, with which it agrees in direction. 

For the case of a circiUar section more definite results can be obtained as follows. If 
we neglect the variations of v and a> over the section, the formulae (1) and (2) give 

* See Cayley, Elliptic FuncHonn, Arts. 72, 77 ; and Maxwell, Lc, 



162^164] ; Speed of a Vortex-Ring 233 

QTy H w6 intrdduoe polar co-ordinatee {s, x) ^ the plAne of the section, 

where a is the radius of the section. Now 

f^ log Mx' = f^ log {^ + *-* - 28^" cos ix - x')}* dx\ 

and this definite integral is known to be equal to 2n log «^ or 2ir log «, according as 9' ^ «. 
Hence, for points within the section, 



^= -»79i 



J' (lOg?^0 .2) ^^r _^^^j- ^i^Sjp _2).W 

= .Woa«{log?j2-f-i^.} 



(6) 



The only variable part of this is the term laxff^; this shews that to our order of 
approanmation the stream-lines within the section are concentric circles, the velocity at a 
distance 8 from the centre being i»8. 

Substituting in Art. 162 ( 1 ) we find 

The last term in Art. 162 (6) is equivalent to 

In our present notation, where k denotes the strength of the whole vortex, this is equal to 
iK^vrJn, Hence the formula for the velocity of translation of the vortex becomes* 

dxn 



t=i^.K-?-i} <'> 



The vortex-ring carries with it a certain body of irrotationaUy moving fluid in its 
career; cf. Art. 166, 2°. According to the formula (7) the velocity of translation of the' 
vortex will be equal to the velocity of the fluid at its centre when wJa=SQf about. The 
accompanying mass will be ring-shaped or not, according as ^sja exceeds or falls short of 
this critical value. 

The ratio of the fluid velocity at the periphery of the vortex to the velocity at the centre 
of the ring is 2aamjK, or wjira. For a =1^9^11 * this \r equal to 32, about. 

164. If we have any number of circular vortex-rings, coaxal or not, the 
motion of any one of these may be conceived as made up of two parts, one 
due to the ring itself, the other due to the influence of the remaining rings. 
The preceding considerations shew that the second part is insignificant 
compared with the first, except when two or more rings approach within 
a very small distance of one another. Hence each ring will move, without 

* This result was given without proof by Sir W. Thomson in an appendix to a translation of 
Hehnholtz' paper, PhU. Mag. (4), t. xxxiii. p. 611 (1867) [Paper9, t. iv. p. 67]. It was verified 
by Hicks, PhiL Trans. A, t. clxxvi. p. 756 (1886); see also Gray, "Notes on Hydrodynamics," 
PhiL Mag, (6), t. xxviii p. 13 (1914). 



234 Vortex Motion [pHAP. vn 

sensible change of shape or size, with nearly nniform velocity in the' 
direction of its rectilinear axis, until it passes within a short distance 
of a second ring.' 

A general notion of the result of the encounter of two rings may, in 
particular cases, be gathered from the result given in Art. 149 (3). Thus, let 
us suppose that we have two circular vortices having the same rectilinear axis. 
If the sense of the rotation be the same for both, the two rings will advance, 
on the whole, in the same direction. One effect of their mutual influence 
will be to increase the radius of the one in front, and to contract the radius 
of the one in the rear. If the radius of the one in front become larger than 
that of the one in the rear, the motion of the former ring will be retarded, 
and that of the latter accelerated. Hence if the conditions as to relative 
size and strength of the two rings be favourable, it may happen that the 
second ring will overtake and pass through the first. The parts played by 
the two rings will then be reversed ; the one which is now in the rear will in 
turn overtake and pass through the other, and so on, the rings alternately 
passing one through the other*. 

If the rotations be opposite, and such that the rings approach one 
another, the mutual influence will be to enlarge the radius of each. If the 
two rings be moreover equal in size and strength, the velocity of approach 
will continually- diminish. In this case the motion at all points of the plane 
which is parallel to the two rings, and half-way between them, is tangential 
to this plane. We may therefore, if we please, regard the plane as a fixed" 
boundary to the fluid on either side, and so obtain the case of a single 
vortex-ring moving directly towards a fixed rigid wall. 

The foregoing remarks are taken from Helmholtz' paper. He adds, 
in conclusion, that the mutual influence of vortex-rings may easily be studied 
experimentally in the case of the (roughly) semicircular rings produced by 
drawing rapidly the point of a spoon for a short space through the surface of 
a liquid, the spots where the vortex-filaments meet the surface being marked 
by dimples. (Cf. Art. 27.) The method of experimental illustration by 
means of smoke-rings f is too well-known to need description here. A beauti- 
ful variation of the experiment consists in forming the rings in water, the 
substance of the vortices being coloured J. 

The motion of a vortex-ring in a fluid limited (whether internally or externally) by a 
fixed spherical surface, in the case where the rectilinear axis of the ring passes through 

* The corresponding case in two dimensions was worked out and illustrated graphically by 
Grobli, Ix. ante p. 218 ; see also Love, " On the Motion of Paired Vortices with a Common Axis,'* 
Proc, Lond. Maih. 8oc. t. zxv. p. 185 (1894). 

t Kousch. **Ueber Ringbildung der Fliissigkeiten/* Pogg. Ann. t. ex. (1800); Tait, Beeent 
Advances in Phyaical Science, London, 1876, c. xii. 

X Reynolds, **0n the Resistance encountered by Vortex Rings &c.,*' Brit, Ass, Sep, 1876; 
Nature, t xiv. p. 477. 



164-165] Mutual Influence of Vortex-Rings 235 

the centre of the sphere, has been investigated by Lewis*, by the method of * images.* 
The following simplified proof is due to Larmor f. The vortex-ring is equivalent (Art. 150) 
to a spherical sheet of double-sources of uniform density, concentric with the fixed sphere. 
The * image' of this sheet will, by Art. 96, be another uniform concentric double-sheet, 
which is, again, equivalent to a vortex-ring coaxal with the fijrst. It easily follows from 
the Art. last cited that the strengths (jc, k') and the radii (or, w') of the vortex-ring and 
its image are connected by the relation 

KOT* + ic'w'* =0. (1) 

The argument obviously appties to the case of a re-entrant vortex of any form, provided 
it he on a sphere concentric with the boundary. 

On the Conditions for Steely Motion. 
165. In steady motion, i,e, when 

dt ' dt "' dt "' 

the equations (2) of Art. 6 may be written 

du , dv , dw , ^ . dCl Idp „. 

Hence, if as in Art. 146 we put 

/=/^ + h»+a (2) 

we have ^ = ^^ "" ^» ^ "" ^^ "" *^^' ^ "^ ^'^ " ^^ (^) 

It f oUows that w 1^' + t; ^ + tr 1^' - 0, 

ox oy oz 

^ dx^'^dy ^^ dz "' 

so that each of the surfaces x' == const, contains both stream-lines and 
vortex-lines. If further 8n denote an element of the normal at any point 
of such a surface, we have 

^-goisinjS, (4) 

where q is the current- velocity, w the vorticity, and j8 the angle between the 
stream-line and the vortez-Une at that point. 

Hence the conditions that a given state of motion of a fluid may be 
a possible state of steady motion are as follows. It must be possible to draw 
in the fluid an inflnite system of surfaces each of which is covered by 

* "On the Images of Vortices in a Spherical Vessel,*' Quart. Jaum, Math. t. xvl p. 338 
(1879). 

t '* Electro-magnetic and other Images in Spheres and Planes," QttarL Joum, Math. t. xxiiL 
p. 94 (1889). 



236 Vortex Motion [chap, vn 

a network of stream-lineg and vortex-lines, and the product jco sin j3 Sn must 
be constant over each such stirface, Sn denoting the length of the normal 
drawn to a consecutive surface of the system*. 

These conditions may also be deduced from the considerations that the 
stream-lines are, in steady motion, the actual paths of the particles, that the 
product of the angular velocity into the cross-section is the same at all points 
of a vortex, and that this product is, for the same vortex, constant with 
regard to the time. 

The theorem that the function x'» defined by (2), is constant over each 
surface of the above kind is an extension of that of Art. 21, where it was 
shewn that x is constant along a stream-line. 

The above conditions are satisfied identically in all cases of irrotational 
motion, provided of course the boundary-conditions be such as are consistent 
with the steady motion. 

In the motion of a Uquid in two dimensions (xy) the product qBn is 
constant along a stream-line; the conditions in question then reduce to this, 
that the vorticity ^ must be constant along each stream-line, or, by 
Art. 59 (5), 

. g+p-/w P) 

where f (iff) is an arbitrary function of ^f- 

This condition is satisfied in all cases of motion in concentric circles about the origin. 
Another obvious solution of (5) is 

Vr =i (Ax* +2Bxy +Cy^) (6) 

in which case the stream-lines are similar and coaxal conic?. The angular velocity at any 
point i3^(A-¥ C), suid is therefore uniform. 

Again, if we put/ (tfr) = - k^, where A; is a constant, and transform to polar co-ordinates 
r, 6, we get 

di^^rW'-^de^^^^-^ ^^^ 

which is satisfied (Art. 101) by ^=CJ,{hr) ^!^Ud (8) 

This gives various solutions consistent with. a fixed circular boundary of radius a, the 
admissible values of h being determined by 

jr,(ifca)=0. :....(9) 

* See a paper *'0n the Conditions for Steady Motion of a Fluid/' Proc, Lond, Math. 8oc, 
t. ix. p. 91 (1878). 

t Cf. Lagrange, Nouv. Mim, de FAead. de Berlin, 1781 [Oeuvres, t. iv. p. 720]; and Stokes, 
''On the Steady Motion of Inoompressible Fluids," Camb, Trans, t. vii (1842) [Papets, t. L 
p. 15]. 



165] Conditions for Steady Motion 237 

Suppose, for example, that in an unlimited ma38 of fluid the stream-function is 

^ =CJi (At) sin ^, (10) 

within the circle r=a, whilst outside this circle we have 



ylr = u(r-'^ame (11) 



These two values of tfr agree for r=a, provided Jj (ka) = 0. Moreover, the tangential velocity 
at this circle will be continuous, provided the two values of 8^/dr are equal, »'.€. if 

If we now impress on everything a velocity U parallel to Ox, we get a species of cylindrical 
vortex travelling with velocity U through a liquid which is at rest at infinity. The 
smallest of the possible values of ik is given by ka/v = 1-2197; the relative stream-lines 
inside the vortex are then given by the lower diagram on p. 280, provided the doUed circle 
be taken as the boundary (r=a). It is easily proved, by Art. 157 (1), that the * impulse* 
of the vortex is represented by 2jrpa*U, 

In the case of motion syinmetrical about an axis (x), we have q . 27rwi8n 
constant along a stream-line, m denoting as in Art. 94 the distance of any 
point from the axis of symmetry. The condition for steady motion then is 
that the ratio ai/t? must be constant along any stream-line. Hence, if ^ be 
the stream-function, we must have, by Art. 161 (2), 



^^&.-i¥.-'"m (.3) 



dx^ 3c5* w dm 
where /(0) denotes an arbitrary function of ^*. 

An interesting example is furnished by Hill*s * Spherical Vortex f.' If we assume 

Vr =i^ar« (a^-r*), (14) 

where r* =a:* +m\ for all points within the sphere r=a, the formula (2) of Art. 161 makes 

so that the condition of steady motion is satisfied. Again it is evident, on reference to 
Arts. 96, 97, that the irrotational flow of a stream with the general velocity - U parallel to 
the axis, past a fixed spherical surface r =a, is given by 



Vr=JC7or«(l-^ (16) 



The two values of y^r agree when r=a; this makes the normal velocity zero on both sides. 
In order that the tangential velocity may be continuous, the values of d^jt/ar must also 
agree. Remembering that or =r sin ^, this gives A = -^U/a^ and therefore 

a> =^Uw/a* (16) 

The sum of the strengths of the vortex-filaments composing the spherical vortex is BUa, 

* This result is due to Stokos, 2.c. 

t '*0n a Spherical Vortex," Phil Trans. A, t. clxxxv. (1894). 



238 



Vortex Motion 



[chap, vn 



The figure shews the stream-lines, both inside and outside the vortex ; they are drawn, 
as usual, for equidistant values of ^. 




If we impress on everything a velocity U parallel to x, we get a spherical vortex 
advancing with constant velocity U through a liquid which is at rest at infinity. 

By the formulae of Art. 162, we readily find that the square of the * mean-radius ' of the 
vortex is fa*, the 'impulse' is 2vpa^U, and the energy is ^{^pa*UK 

As explained in Art. 146, it is quite unnecessary to calculate formulae for the pressure, 
in order to assure ourselves that this is* continuous at the surface of the vortex. The con- 
tinuity of the pressure is already secured by the continuity of the velocity, and the constancy 
of the circulation in any moving circuit. 

166. As already stated, the theory of vortex motion was originated 
by Helmholtz in 1858. It acquired additional interest when, in 1867, 
Kelvin suggested* the theory of vortex atoms. As a physical theory, this 
lies outside our province, but it has given rise to a great number of interesting 
investigations, to which some reference should be made. We may mention 
the investigations as to the stability and the periods of vibration of recti- 
linear f and annular]: vortices; the similar investigations relating to hollow 
vortices (where the rotationally moving core is replaced by a vacuum §) ; and 
the calculations of the forms of boundary of a hollow vortex which are con- 
sistent with steady motion ||. A summary of some of the leading results has 
been given by Love^f. 

* Lc, ante p. 216. 

t Sir W. Thomson, Ic, ante p. 224. 

X J. J. Thomaon, l.c. ante p. 209; Dyson, Phil. Trans. A, t. clxxxiv. p. 1041 (1893). 

§ Sir W. Thomson, l.c.; Hicks, "On the Steady Motion and the Small Vibrations of a 
Hollow Vortex," PhU. Trans. 1884; Pocklington, "The Complete System of the Periods of 
a Hollow Vortex Ring," Phil Trans. A, t. clxxxvi p. 603 (1896); Carslaw, "The Fluted 
Vibrations of a Circular Vortex-Ring with a HoUow Core," Proc. Land. Math. Soc. t. xxviii. 
p. 97 (1896). 

ii Hicks, l.c. ; Pocklington, " Hollow Straight Vortices," Camb. Proc. t. viiL p. 178 (1894). 

II l.c. anUp. 183. 



16&-167] HilTs Spherical Vortex 239 

ClebscVs Transformation. 

167. Another matter of some interest, which can however only be briefly 
touched upon, is Clebsch's transformation of the hydrodynamical equations*. 

It is easily seen that the component velocities at any one instant can be expressed in 
the forms 

where <^, X, la are functions of z, y, z, provided the component rotations can be put in the 

forms 

8(X^) d{\,^) 8(X,/x) 

^~d(y.zy '^''d(z,xy ^-d(x,y) ^^^ 

Now if the differential equations of the vortex -lines, viz. 

dx _dy _dz 

y"7"T ^^^ 

be supposed integrated in the form 

a =const., /3 =const (4) 

where a, /3 are functions of x, y, z, we must have 

^ p 8(a,g ) _p 8(a, ff) ^_p 8(a, ff) 

^-^8(y,z)' ''-^8(«,a;)' '■~^8(a:,y) ^^^ 

where P is some function of x, y, zf* Substituting these expressions in the identity 

oa? cy 02 

wefind 8 (P, a, ^) ^Q 

8 (a?, y, 2) ' ' • 

which shews that P is of the form / (a, /3). If X, /ui be any two functions of a, /3, we have 

3(y.*) S(a,^) 8(y.z)' » ' » • 

and the eqtiations (6) will therefore reduce to the form (2), provided X, m be choeen so that 

lgrg=/(-^) (') 

which can obviously be satisfied in an infinity of ways. 

It is evident from (2) that the intersections of the surfaces X = const., /i = const, are the 
vortex-lines. This suggests that the functions X, /jl which occur in (1) may be supposed to 
vary continuously with t in such a way that the surfaces in question move with the fluid $. 
Various analytical proofis of the possibility of this have been given ; the simplest, perhaps, 
is by means of the equations (2) of Art. 16, which give (as in Art. 17) 

udx +vdy+wdz=UQda +VQdb +WQdc -dx (8) 

It has been proved that we may assume, initially, 

ti^da+VQdb+WQdc^ -(2</>o+X(2fu (9) 

* "Ueber eine aUgemeine Transformation d. hydrodynamlBchen Qleichungen," Crelle, t. liv. 
(1857) and t. IvL (1860). See also Hill, QtuirL Joum, Math, t. xvii. (1881), and Camb. Trans. 
t. xiv. (1883). 

t GL Forsyth, DifferenJtidl Egnations, Art. 174. 

X It must not be overlooked that on account of the insuflGioient detemunacy of \, fi these 
functions may vary continuously with t without relating always to the same particles of fluid. 



240 Vortex Motion [chap, vn 

Hence, considering space- variations at time t^ we shall have 

■ * 

udx .+vdy +wdz = -d^i +Xrff*, (10) 

f . . . .' 

where^ ^ s^, +x> aad X, fi have the same values as in (9), but are now expressed in terms 

of X, y, Zy t. Since, in the 'Lagrangian* method the independent space- variables relate to 

the individual particles, this proves the theorem. 

On this understanding the equations of motion can be integrated, provided the 
extraneous forces have a potential, and that p is a function of p only. We have 

^w ft J. « 8tt / 5X d\ dK\ 9u / du. du 3u\ 9X 

~dx\~dt'^''dt)'^ Dldx~ Dtdx' ^"' 

and therefore, on the present Assumption that D\/Dt =0, Dn/Dl =0, 

J^^.1^.0=|-X| (12) 

by Art. 146 (5), (6). An arbitrary function of t is here supposed incorporated in d^/8^ 
If the above condition be not imposed on X, /x, we have, writing 

^=/?-i^-°-t-4 (13) 

Dtdx''lH^~'dx' Dtdy''Dtdy~ dy* Dt dz Dt dz" Zz'"'^^ 

Hence Tr^^=^ d^) 

shewing that H is of the form / (X, ^, t) ; and 

DK_ m Dfi_dH 

IH~~d,i' Dt~dX (^"' 



CHAPTER VIII 



TIDAL WAVES 



168. One of the most interesting and successful applications of hydro- 
dynamical theory is to the small oscillations, under gravity, of a liquid having 
a free surface. In certain cases, which are somewhat special as regards the 
theory, but very important from a practical point of view, these oscillations 
may combine to form progressive waves travelling with (to a first approxi- 
mation) no change of form over the surface. 

The term ^ tidal,' as applied to waves, has been used in various senses, but 
it seems most natural to confine it to gravitational oscillations possessing the 
characteristic feature of the oceanic tides produced by the action of the sun 
and moon. We have therefore ventured to place it at the head of this 
Chapter, as descriptive of waves in which the motion of the fluid is mainly 
horizontal, and therefore (as will appear) sensibly the same for all particles 
in a vertical line. This latter circumstance greatly simplifies the theory. 

It will be convenient to recapitulate, in the first place, some points in the 
general theory of small oscillations which will receive constant exemplification 
in the investigations which follow*. The theory has reference in the first 
instance to a system of finite freedom, but the results, when properly inter- 
preted, hold good without this restriction f. 

Let ?i, ?2, . . . Jn be n generalized co-ordinates serving to specify the con- 
figuration of a dynamical system, and let them be so chosen as to vanish in 
the configuration of equilibrium. The kinetic energy T will be a homogeneous ^ 
quadratic function of the generalized velocities qi, q^f > • » ^m ^7 

2T = aiiji* + «2292* + • . • + 2aijj^ig2 + • • • > (1) 

where the coefficients are in general functions of the co-ordinates 9i, 929 • - • 9n > 
but may in the application to small motions be supposed constant, and to 
have the values corresponding to 9x> ?29 • • • 9n "= 0. Again, if (as we shall 

* For a fuller account of the general theory see Thomson snd Tait, Arts. 337, . . . ; Rayleigh, 
Tlieory of Sound, c. iv. ; Routh, Elementary Rigid Dynamics (6th ed.), London, 1807» c. ix. ; 
Whittaker, Analytical Dynamics, o. vii 

t The steps by which a rigorous transition can be made to the case of infinite freedom have 
been investigated by Hilbert, Q&L Nachr. 1004, p. 40. 

L. H. 16 



242 Tidal Waves [chap, vin 

suppose) the system is 'conservative/ the potential energy F of a small 
displacement is a homogeneous quadratic function of the component 
displacements q^^ 9if -- - 9n* ^^^ X^^ the same understanding) constant 
coefficients, say 

2V = CuJx* H- Cjajj* + . . . + 2ci,grigr, + (2) 

By a real* linear transformation of the co-ordinates 9i, 92, . . . 9n ^^ ^ 
possible to reduce T and V simultaneously to sums of squares; the new 
variables thus introduced are called the * normal co-ordinates' of the system. 
In terms of these we have 

2r = aigi« + a,g,* + ... H-a„g««, (3) 

2F = CiJi« + c,ft« + ... -hCnqn^ (4) 

The coefficients a^y a^, . » . a^ are called the ^principal coefficients of inertia' ; 
they are necessarily positive. The coefficients Ci, c^^ . , . c^ may be called the 
' principal coefficients of stability ' ; they are all positive when the undisturbed 
configuration is stable. 

When given extraneous forces act on the system, the work done by these 
during an arbitrary infinitesimal displacement A^^, Aq^, . . . Ag^ may be 
expressed in the form 

GiA?x + QaAj,+ ... +Q„A?„. (5) 

The coefficients Qi, Q^, ... Qn a*re then called the 'normal components of 
disturbing force.' 

In the appUcation to infinitely small motions Lagrange's equations 
take the form 

«1.4'l + «2r?2 + • • • -^^Itqi + C2f?« + ...=* Or (7) 

or, in the case of normal co-ordinates, 

«r9r + Crqr = Qr (8) 

•It is easily seen from this that the dynamical characteristics of the normal 
co-ordinates are (F) that an impulse of any normal type produces an initial 
motion of that type only, and (2°) that a steady disturbing force of any type 
maintains a displacement of that type only. 

To obtain the free motions of the system we put Qr = 0. Solving (8), 
we find 

qr = Ar COS (art + €r\ (9) 



where a 



'-©• ™ 



* The algebraic proof of this involves the assnmption that one at least of the functions T, V 
is essentially positive. In the present case T of conrse fulfils this condition. 



168] Small Oscillations 243 

and A^t e^ are arbitrary constants*. Hence a mode of free motion is possible 
in which any normal co*ordinate q^ varies alone, and the motion of any particle 
of the system, since it depends linearly on q^t will be simple-harmonic, of 
period 2^/(7^; moreover the particles will pass simultaneously through their 
equilibrium positions. The several modes of this character are called the 
'normal modes' of vibration of the system; their number is equal to that of 
the degrees of freedom, and any free motion whatever of the system may be 
obtained from them by superposition, with a proper choice of the * amplitudes' 
(-4^) and * epochs ' (c^). 

In certain cases, viz. when two or more of the free periods (2w/a) of the 
system are equal, the normal co-ordinates are to a certain extent indeterminate, 
i,e, they can be chosen in an infinite number of ways. An instance of this is 
the spherical pendulum. Other examples will present themselves later ; see 
Arts. 191, 200. 

If two (or more) normal modes have the same period, then by compounding 
them, with arbitrary amplitudes and epochs, we obtain a small oscillation 
in which the motion of each particle is the resultant of simple-harmonic 
vibrations in different directions, and is therefore, in general, elliptic-harmonic, 
with the same period. This is exemplified in the spherical pendulum; an 
important instance in our own subject is that of progressive waves in deep 
water (Chapter ix.). 

If any of the coefficients of stability (c^) be negative, the value of a^ is 
a pure imaginary. The circular function in (9) is then replaced by real ex- 
ponentials, and an arbitrary displacement will in general increase until the 
assumptions on which the approximate equation (8) is based become untenable. 
The undisturbed configuration is then reckoned as unstable. The necessary 
and sufficient condition of stability (in the present sense) b that the potential 
energy V should be a minimum in the configuration of equilibrium. 

To find the effect of disturbing forces, it is sufficient to consider the case 
where Q^ varies as a simple-harmonic function of the time, say 

Q^ = 0^ cos (a« -f €), (11) 

where the value of a is now prescribed. Not only is this the most interesting 
case in itself, but we know from Fourier's Theorem that, whatever the law of 
variation of Q^ with the time, it can be expressed by a series of terms such as 
(11). A particular integral of (8) is then 

g^° r ^'^a <^Qs(<^ + ^) (12) 

* The ratio <r/2T measures the 'frequency' of the oscillation. It is convenient to have a 
name for the quantity a itself; the term * speed* has been used in this sense by Kelvin and 
0. H. Darwin in their researches on the Tides. 

16—2 



244 Tidal Waves [chap, vra 

This represents the 'forced oscillation' due to the periodic force Q^. In it 
the motion of every particle is simple-harmonic, of the prescribed period 
2^/<7, and the extreme displacements coincide in time with the maxima and 
minima of the force. 

A constant force equal to the instantaneous value of the actual force (11) 

would maintain a displacement 

n 
5^ « ^ cos (a< + €), (13) 

the same, of course, as if the inertia-coefficient a^ were null. Hence (12) may 
be written 

where g^ has the value (10). This very useful formula enables us to write 
down the effect of a periodic force when we know that of a steady force of the 
same type. It is to be noticed that q^ and Q^ have the same or opposite 
phases according as a $ a^, that is, according as the period of the disturbing 
force is greater or less than the free period. A simple example of this is 
furnished, by a simple pendulum acted on by a periodic horizontal force. 
Other important illustrations will present themselves in the theory of the 
tides*. 

When a is very great in comparison with a,., the formula (12) becomes 

Q 

?r = - -2^ cos (<rf H- €) ; (15) 

the displacement is now always in the opposite phase to the force, and 
depends only on the inertia of the system. 

If the period of the impressed force be nearly equal to that of the normal 
mode of order r, the amplitude of the forced oscillation, as given by (14), is 
very great compared with g^. In the case of exact equality, the solution (12) 
fails, and must be replaced by 

gr^ = 5e sin (a« + e), (16) 

where, as is verified immediately on substitution, B = C,./2<7a^. This gives 
an oscillation of continually increasing amplitude, and can therefore only 
be accepted as a representation of the initial stages of the disturbance. 

Another very important property of the normal modes may be noticed. If by the 
introduction of constraints the system be compelled to oscillate in any other prescribed 
manner, the configuration at any instant c€Ui be specified by one variable, which we will 
denote by 6, In terms of this we shall have 

qr=Br6, 

* Cf. T. Young, "A Theory of Tides," NichoUorCa Journal, t. xzzv. (1813) [MidceUaneow 
Works, London, 1864, t. u. p. 262]. 



168] The(yry of Normal Modes 245 

where the quantities Br are certain oonstants. This makes 

2T=(Bi«ai +B,«a, + . . . +B»«an)^, (17) 

2F=(Bi«Ci-fB,*c,+ ... +B»«0^ (18) 

If ^ a cos (crt +€), the constancy of the energy {T + V) requires 

Hence o-' is intermediate in value between the greatest and least of the quantities Crja^ ; 
in other words, the frequency of the constrained oscillation is intermediate between the 
greatest and least frequencies corresponding to the normal modes of the system. In par- 
ticular, when a system is modified by the introduction of a constraint, the frequency of 
the slowest natural oscillation is irkcreased. 

Moreover, if the constrained type differ but slightly from a normal type (r), cr' will 
differ from (v/Or by a small quantity of ike second order. This gives a valuable method of 
estimating approximately the frequency in cases where the normal types cannot be 
accurately determined*. 

It may further be shewn that in the case of a partial constraint, which merely reduces 
the degree of freedom from n to n - 1, the periods of the modified system separate those of 
the original onef . 

It is of some interest in the present connection to recall a remark made by Lagrange in 
the M&aniqiu AnalyiiqueX to the effect that if in the equations of type (7), where the 
co-ordinates are not assumed to be normal, we put Qr=0, and assume 

qr^Ar^\ ...(20) 

the resulting equations are identical with those which determine the stationary values 
( - X*) of the expression 

^1-^1 "^ ^-^1 + " » + 2ci2AiA^ + . . . ^ .gj. 
^1^1*+ ^^2*+ •'• + ^Oiji^i^, + 

Since T is essentially positive the denominator cannot vanish, and the expression has 
therefore a minimum value. 

It is moreover possible, starting from this property, to construct a proof that the n 
values of X' are all real§. They are obviously all negative if F be essentially positive. 

Rayleigh*s theorem is also closely related to the Hamiltonian formula (19) of Art. 135, 
as we may see by assuming 

qr = ArBUKrt (22) 

and taking ^^ = 0, ^ = 2ir/<r. 

The modifications which are introduced into the theory of small oscillations 
by the consideration of viscous forces will be noticed in Chapter xi. 

* Rayleigh, "Some General Theorems relating to Vibrations," Proc, Lond, Math. 8oc. t. iv. 
p. 367 (1874) [Paperd, t. L p. 170], and Theory of Sound, o. iv. The method is elaborated by 
Bitz, Joum.f4kr Math., t. cxxxv. p. 1 (1908), and Ann. der Physik, t. xxviil (1909) [Gesammelte 
Werke, Paris, 1911, pp. 192, 2S6]. 

t Routh, Elementary Rigid Dynamics, Art. 67 ,- Rayleigh, Theory of Sound (2nd ed.). Art. 92 a ; 
Wbittaker, Analytical Dynamics, Art. 81. 

X Oeuvres, t. xi, p. 380. 

§ See Poinoar^, Jowm. de Math. (6), t. ii. p. 83 (1896). 



246 Tidal Waves [chap, vm 



Long Waves in Canals. 

169. Proceeding now to the special problem of this Chapter, let us begin 
with the case of waves travelling along a straight canal, with horizontal bed, 
and parallel vertical sides. Let the axis of x be parallel to the length of the 
canal, that of y vertical and upwards, and let us suppose that the motion 
takes place in these two dimensions x, y. Let the ordinate of the free surf ace, 
corresponding to the abscissa x, at time f, be denoted by y^ + % where yo ^ 
the ordinate in the undisturbed state. 

As already indicated, we shall assume in all the investigations of this 
Chapter that the vertical acceleration of the fluid particles may be neglected, 
or, more precisely, that the pressure at any point (a;, y) is sensibly equal to 
the statical pressure due to the depth below the free surface, viz. 

' . p-Po = 9p(yo + v-y)y (1) 

where p^ is the (uniform) external pressure. 

H«°c« l=^''i (2) 

This is independent of y, so that the horizontal acceleration is the same for 
all particles in a plane perpendicular to x. It follows that all particles which 
once lie in such a plane always do so ; in other words, the horizontal velocity 
uiB a, function of x and t only. 

The equation of horizontal motion, viz. 

du du ^ 1 dp 
dt dx" pdx' 

is further simplified in the case of infinitely small motions by the omission of 
the term udu/dxy which is of the second order, so that 

Now let f = Jw(ft; 

i.e. ^ is the time-integral of the displacement past the plane x, up to the 
time t In the case of smaU motions this will, to the first order of small 
quantities, be equal to the displacement of the particles which originally 
occupied that plane, or again to that of the particles which actually occupy it 
at time t. The equation (3) may now be written 

ar' = -^g w 



169] Waves in Uniform Canal 247 

The equation of contmuity may be found by calculating the volome of 
fluid which has, up to time ty entered the space bounded by the planes x and 
X + Sx; thus, if A be the depth and b the breadth of the canal, 

- g^ (f A6) Sx = -ribhx, 

♦ 

The same result comes from the ordinary form of the equation of con- 
tinuity, viz. 

i+i=» <«) 

"-/Is^-^S <" 

if the origin be (for the moment) taken in the bottom of the canal. 
This formula is of interest as shewing that the vertical velocity of any 
particle is simply proportional to its height above the bottom. At the 
free surface we have y = A + 17, v = 3iy/%, whence (neglecting a product of 
small quantities) 

di^^^d^t ^^^ 

From this (5) follows by integration with respect to L 
Eliminating 17 between (4) and (5), we obtain 

a? = ^*^« <^) 

The elimination of ^ gives an equation of the same form, viz. 

^-^"^ « 

The above investigation can readily be extended to the case of a 
imiform canal of any form of section*. If the sectional area of the un- 
disturbed fluid be Sy and the breadth at the free surface 6, the equation of 
continuity is 

- 4 (^5) 8x = 7,68x, (11) 

whence ly = — A^, (12) 

as before, provided h =■ S/b, i.e. h now denotes the mean depth of the canaL 
The dynamical equation (4) is of course unaltered. 

* Kelland, Trans. JR. 8. Sdin. t. xiv. (1839). 



248 Tidal Waves [chap, vm 

170. The equation (9) is of a well-known type which occurs in several 
physical problems, e.g. the transverse vibrations of strings, and the motion of 
sound-waves in one dimension. 

To integrate it, let us write, for shortness, 

c^^gK (13) 

and a? — c^ = x^^ x •\- ct ^= x^. 

In terms of x^ and x^ as independent variables, the equation takes the form 

dx-i dx^ 

The complete solution is therefore 

^ = F(x^ct) +f(x + ct), (14) 

where F, / are arbitrary functions. 

The corresponding values of the particle-velocity and of the surface- 
elevation are given by 



"^ I..- (15) 

5=-F(a5-ce)-/(a? + c<).J 



The interpretation of these results is simple. Take first the motion 
represented by the first term in (14), alone. Since F {x-^ ct) is imaltered 
when t and x are increased by r and cr, respectively, it is plain that the dis- 
turbance which existed at the point x at time t has been transferred at time 
< + T to the point x -\- cr. Hence the disturbance advances unchanged with a 
constant velocity c in space. In other words we have a * progressive wave' 
travelling with constant velocity c in the direction of a-positive. In the same 
way the second term of (14) represents a progressive wave travelling with 
velocity c in the direction of x-negative. And it appears, since (14) is the 
complete solution of (9), that any motion whatever of the fluid, which is 
subject to the conditions laid down in the preceding Art., may be regarded as 
made up of waves of these two kinds. 

The velocity (c) of propagation is, by (13), that *due to' half the depth of 
the undisturbed fluid*. 

The foUowing table, giving in round numbers the velocity of wave-propagation for 
various depths, will be of interest later in connection with the theory of the tides. 

The last column gives the time a wave would take to travel over a distance equal to 
the earth*8 circumference (2«ra). In order that a 'long* wave should traverse this distance 
in 24 hours, the depth would have to be about 14 miles. It must be borne in mind that 

* Lagrange, Nouv. nUm, de VAcad, de Berlin, 1781 [OeuvreSf t. i p. 747]. 



170-171] 



Wave- Velocity 



249 



these numerioal results are only applicable to waveis satisfying the conditions above 
postulated. The meaning of these conditions will be examined more particularly in 
Art 172. 



h 


c 


c 


2ira/e 


(feet) 


(feet per sec.) 


(sea-miles per hour) 


(hours) 


312i 


100 


60 


360 


1250 


200 


120 


180 


5000 


400 


240 


90 


11250* 


600 


360 


60 


20000 


800 


480 


45 



171. To trace the effect of an arbitrary initial disturbance, let us suppose 
that when ^ = we have 

^=^(a;), l^'^ix) (16) 

The functions F'^f which occur in (15) are then given by 

F'{x) = -l{4>(x) + ^{x)},) 

fix)^ H<f> i<») - 'I' im ^ ' 

Hence if we draw the curvea y = i)i, y = rj^, where 

Vi = ih{^{x)+<f,{x)},\ 

rit = hf^{^l»(x)-4>(x)),\ ^""^ 

the form of the wave-profile at any subsequent instant t is found by displacing 
these curves parallel to x, through spaces ± ct, respectively, and adding (alge- 
braically) the ordinates. If, for example, the original disturbance be confined 
to a length I of the axis of a;, then after a time 2/2c it will have broken up 
into two progressive waves of length I, travelling in opposite directions. 

In the particular case where in the initial state f = 0, and therefore 
<l> {x) = 0, we have lyi = ^a 5 *t® elevation in each of the derived waves is then 
exactly half what it was, at corresponding points, in the original disturbance. 

It appears from (16) and (17) that if the initial disturbance be such that 
^ = ± Tj/h . c, the motion will consist of a wave system travelling in one 
direction only, since one or other of the functions F' and/' is then zero. 

It is easy to trace the motion of a surface-particle as a progressive wave 
of either kind passes it. Suppose, for example, that 

i = F{x-ct), (19) 

and therefore f = c? (20) 

* This Jb probably comparable in order of magnitude with the mean depth of the ocean. 



250 TidcU Waves [ohap. vm 

The particle is at rest until it is reached by the wave ; it then moves forward 
with a velocity proportional at each instant to the elevation above the mean 
level, the velocity being in fact less than the wave- velocity c, in the ratio of 
the surface-elevation to the depth of the water. The total displacement at 
any time is given by 



^ ~ hi ^^^' 



This integral measures the volume, per unit breadth of the canal, of the 
portion of the wave which has up to the instant in question passed the 
particle. Finally, when the wave has passed away, the particle is left at rest 
in advance of its original position at a distance equal to the total volume of 
the elevated water divided by the sectional area of the canal. 

172. We can now examine under what circumstances the solution ex- 
pressed by (14) will be consistent with the assumptions made provisionally 
in Art. 169. 

The exact equation of vertical motion, viz. 

Dv dp 

gives, on integration with respect to y, 

fVo+n Df) 

p-'Po = 9p(yo + v-y)-pj -j^^y (2i) 

This may be replaced by the approximate equation (1), provided j3 (A + 17) be 
small compared with ^, where j3 denotes the maximum vertical acceleration. 
Now in a progressive wave, if A denote the distance between two consecutive 
nodes (i.e. points at which the wave-profile meets the undisturbed level), the 
time which the corresponding portion of the wave takes to pass a particle is 
A/c, and therefore the vertical velocity will be of the order lyc/A*, and the 
vertical acceleration of the order r)C*IX\ where r) is the maximum elevation 
(or depression). Hence the neglect of the vertical acceleration is justified, 
•provided A*/A* is a small quantity. 

Waves whose slope is gradual, and whose length A is large compared with 
the depth h of the fluid, are called 'long waves.' 

Again, the restriction to infinitely small motions, made in equation (3), 
consisted in neglecting udu/dx in comparison with du/dt. In a progressive 
wave we have du/dt — ± cdu/dx ; so that u must be small compared with c, and 
therefore, by (20), t) must be small compared with h. It is to be observed 
that this condition is altogether distinct from the former one, which may be 
legitimate in cases where the motion cannot be regarded as infinitely small. 
See Art. 187. 

* Hence, comparing with (20), we see that the ratio of the maximum vertical to the maximum 
horizontal velocity is of the order h/\. 



171-173] Airy's Method 251 

The preceding conditions will of course be satisfied in the general ciase 
represented by equation (14), provided they are satisfied for each of the two 
progressive waves into which the disturbance can be analysed. 

173. There is another, although on the whole a less convenient, method 
of investigating the motion of Uong' waves, in which the Lagrangian plan is 
adopted, of making the co-ordinates refer to ttie individual particles of the 
fluid. For simplicity, we will consider only the case of a canal of rectangular 
section*. The fundamental assumption that the vertical acceleration may be 
neglected impUes as before that the horizontal motion of all particles in a 
plane perpendicidar to the length of the canal will be the same. We there- 
fore denote by x + f the abscissa at time t of the plane of particles whose 
undisturbed abscissa is x. If 77 denote the elevation of the free surface, in 
this plane, the equation of motion of unit breadth of a stratum whose thick- 
ness (in the undisturbed state) is hx will be 



^*^S=-|^^ (*+'')' 



where the factor {dp/dx) . 8a: represents the pressure-difEerence for any two 
opposite particles x and x + 8a? on the two faces of the stratum, while the 
factor h-\- 7) represents the area of the stratum. Since we assume that the 
pressure about any particle depends only on its depth below the free surface 
we may write 

dx^^^dx' 
so that our dynamical equation is 

^f--*('+'»)i <•) 

The equation of continuity is obtained by equating the volumes of a stratum, 
consisting of the same particles, in the disturbed and undisturbed conditions 
respectively, viz. 

(&x + 1^ 8x^ (h-^Tj)^ h8x, 

'+i-(»+i)"' <^) 

Between equations (1) and (2) we may eHminate either t] or (; the result in 
terms of f is the simpler, being 



or 



a«f dx^ 



a# = ^*77~5p ^^^ 



* Airy, Encye. Mdrop. "Tides and Waves," Art. 192 (1845); see also Stokes, "On Waves," 
Cawh, and Dub, Maih. Joum. t. iv. (1849) [Papers, t. ii. p. 222]. The case of a canal with sloping 
sides has been treated by McCowan, **0n the Theory of Long Waves. . .," PML Jliag. (5), t. xxxv. 
p. 250 (1892). 



252 Tidal Waves [chap, vm 

This is the general equation of Uong' waves in a unifonn canal with vertical 
sides*. 

So far the only assumption is that the vertical acceleration of the particles 
may be neglected in calcidating the pressure. If we now assume, in addition, 
that rijh is a small quantity, the equations (2) and (3) reduce to 

^--^t • (*) 

*^d W^^^d^ <^) 

The elevation tj also satisfies the equation 

S=^*gS (^) 

These are in conformity with our previous results; for the smaUness of 
d^/dx means that the relative displacement of any two particles is never more 
than a minute fraction of the distance between them, so that it is (to a first 
approximation) now immaterial whether the variable x be supposed to refer 
to a plane fixed in space, or to one moving with the fluid. 

174. The potential energy of a wave, or system of waves, due to the 
elevation or depression of the fluid above or below the mean level is, per unit 
breadth, gp ilydxdy, where the integration with respect to y is to be taken 
between the limits and 77, and that with respect to x over the whole length 
of the waves. Effecting the former integration, we get 

\9Ph*dx (1) 

The kinetic energy is \ph /|*(fo (2) 

In a system of waves travelling in one direction only we have 

so that the expressions (1) and (2) are equal; or the total energy is half 
potential, and half kinetic. 

This result may be obtained in a more general manner, as foUowsf* Any 
progressive wave may be conceived as having been originated by the spUtting 
up, into two waves travelling in opposite directions, of an initial disturbance 
in which the particle- velocity was everywhere zero, and the energy therefore 
wholly potential. It appears from Art. 171 that the two derived waves are 
symmetrical in every respect, so that each must contain half the original 
store of energy. Since, however, the elevation at corresponding points is for 
each derived wave exactly half that of the original disturbance, the potential 

♦ Airy, l.c* 

t Rayleigh, "On Waves," PhU, Mag. (5), t. i. p. 257 (1876) [Papers, t. L p. 251]. 



173-176] Energy 253 

energy of each will by (1) be one-fourth of the original store. The remaining 
(kinetic) part of the energy of each derived wave must therefore also be one- 
fourth of the original quantity. 

175. If in any case of waves travelUng in one direction only, without 
change of form, we impress on the whole mass a velocity equal and opposite 
to that of propagation, the motion becomes steady ^ whilst the forces acting on 
any particle remain the same as before. With the help of this artifice, the 
laws of wave-propagation can be investigated with great ease*. Thus, in the 
present case we shall have, by Art. 22 (4), at the free surface, 

^ = const. - 5^ (A + ^) - \q\ (1) 

where q is the velocity. If the slope of the wave-profile be everywhere 
gradual, and the depth h small compared with the length of a wave, the 
horizontal velocity may be taken to be uniform throughout the depth, and 
approximately equal to q. Hence the equation of continuity is 

J (A + ^) = cA, 

c being the velocity, in the steady motion, at places where the depth of the 
stream is uniform and equal to h. Substituting for q in (1), we have 

? = con8t.-^A(l+|)-ic«(l + 5)-\ 

Hence if ry/A be small, the condition for a free surface, viz. p = const., is 
satisfied approximately, provided 

which agrees with our former result. 

176. It appears from the hnearity of our equations that, in the case of 
sufficiently low waves, any number of independent solutions may be super- 
posed. For example, having given a wave of any form travelling in one 
direction, if we superpose its image in the plane x = 0, travelHng in the 
opposite direction, it is obvious that in the resulting motion the horizontal 
velocity will vanish at the origin, and the circumstances are therefore the 
same as if there were a fixed barrier at this point. We can thus understand 
the reflexion of a wave at a barrier; the elevations and depressions are 
reflected unchanged, whilst the horizontal velocity is reversed. The same 
results follow from the formula 

^^F{ct-x)-F(ct^x)y (1) 

which is evidently the most general value of ^ subject to the condition that 
f = f or a; = 0. 

* Rayleigh, Z.c. 



I 

t 



f 



I 



264 Tided Waves [chap, vm 

We can farther inyestigate without much difficulty the partial reflexion of a wave at a 
point where there is an abrupt change in the section of the canaL Taking the origin at 
the point in question, we may write, for the negative side, 

,.=i^('-£)+/(*-^5). «.=J^('-S-J/('-^|) (2) 

and for the positive side 

'«=*('-6' "^^JK'-O <'> 

where the function F represents the original wave, and /, ^ the reflected and transmitted 
portions respectively. The constancy of mass requires that at the point a; =0 we should 
have hihitii =bj^u^, where bi, b^ are the breadths at the surface, and h^, k^ are the mean 
depths. We must also have at the same point 171=172, ^^ account of the continuity of 
pressure*. These conditions give 

^ {F «) -/ («)} =^ <f> (t). F (*) +/ (0 =4, (0. 

We thence find that the ratios of the elevations in corresponding parts of the reflected and 
incident waves, and of the transmitted and incident waves, are 

F 6iCi+6gCj' F bjCj+bfy' ^^ 

respectively. The reader may easily verify that the energy contained in the reflected and 
transmitted waves la equal to that of the original incident wave. 

177. Our investigations, so far, relate to cases oifree waves. When, in 
addition to gravity, small disturbing forces X, Y act on the fluid, the equation 
of motion is obtained as follows. 

We assume that within distances comparable with the depth h these 
forces vary only by a small fraction of their total value. On this under- 
standing we have, in place of Art. 169 (1), 

^-^ = {9-Y){y, + r,-y) (1) 

and therefore i || = (<; - 7) g - (y, + , - y) g. 

The last term may be neglected for the reason just stated, and if we 
further neglect the product of the small quantities Y and drj/dx, the equation 
reduces to 

-?? = fl^ (2) 

pdx^dx' ^^' 

* It will be understood that the problem admits only of an approximate treatment, on account 
of the rapid change in the character of the motion near the point of discontinuity. The nature 
of the approximation implied in the above assumptions will become more evident if we suppose 
the suffixes to refer to two sections S^ and 8^, one on each side of the origin 0, at distances from 
which, though very small compared with the wave-length, are yet moderate multiples of the 
transverse dimensions of the canaL The motion of the fluid will be sensibly uniform over each 
of these sections, and parallel to the length. The condition in the text then expresses that there 
is no sensible change of level between Si and 3f, 



176-178] Disturbing Fwce» 265 

as before. The equation of hojrizontal motion then takes tke form 

^^-'S+^. <») 

where X may be regarded as a function of x and t only. The equation of 
continuity has the same form as in Art. 169, viz. 

^=-*i- ••• w 

Hence, on elimination of i\, 

w'^^w^^^ (^) 



• 



178. The oscillations of water in a canal of uniform section, closed at 
both ends, may, as in the corresponding problem of Acoustics, be obtained by 
superposition of progressive waves travelling in opposite directions. It is 
more instructive, however, with a view to subsequent more difficult investi- 
gations, to treat the problem as an example of the general theory sketched in 
Art. 168. 

We have to determine ^so as to satisfy 

8? "^S^^"^^' (^^ 

together with the terminal conditions that ^ = f or a; = and x = I, say. 
To find the free oscillations we put Z «= 0, and assume that 

^ OC cos {(ft + €), 

where a is to be found. On substitution we obtain 

g+^f-» <^) 

whence, omitting the time-factor, 

f = 4 sm V B cos — . 

The terminal conditions give £ ^ 0, and 

aljc — nr, (3) 

where r is integral. Hence the normal mode of order r is given by 

* . . nrx (met , \ ... 

i = A^ sm -y cos [-Y- + €^j, (4) 

where the amplitude Af and epoch €«. are arbilarary. 

In the slowest oscillation (r = 1), the water sways to and fro, heaping 
itself up alternately at the two ends, and there is a node at the middle 
(x = \l). The period (2I/c) is equal to the time a progressive wave would 
take to traverse twice the length of the canaL 



256 Tidal Waves [chap, vin 

The periods of the higher modes are respectively J, J, i, ... of this, but 
it must be remembered, in this and in other similar problems, that our theory 
ceases to be appUcable when the length Ijr of a semi-undulation becomes 
comparable with the depth h. 

On comparison with the general theory of Art. 168, it appears that the 
normal co-ordinates of the present system are quantities ji, jj, ... q^ such 
that when the system is displaced according to any one of them, say g^, we 
have 

> tttx 

f «y^sm-y-; 

and we infer that the most general displacement of wliich the system is 
capable (subject to the conditions presupposed) is given by 

^ = Sy^ sin -y , (5) 

where ji, q%, >>. qn are arbitrary. This is in accordance with Fourier's 
Theorem. 

When expressed in terms of the normal velocities and the normal co-ordi- 
nates, the expressions for T and Y must reduce to sums of squares. This is 
easily verified, in the present case, from the formida (6). Thus if S denote 
the sectional area of the canal, we find 

2T = pSJ i^dx = So^^2, 2V=gp^l yj^dx^ Sc^jA • • • (6) 

where o^ = ipSl, c^ = ir^7r^gphS/l (7) 

It is to be noted that, on the present reckoning, the coefficients of stability 
(Cr) increase with the depth. 

Conversely, if we assume from Fourier's Theorem that (5) is a sufficiently 
general expression for the value of ^ at any instant, the calculation just 
indicated shews that the coefficients qr are the normal co-ordinates ; and the 
frequencies can then be found from the general formula (10) of Art. 168; viz. 
we have 

^r = (CrK)* = rir (5rA)*/Z (8) 

in agreement with (3). 

179. As an example of forced waves we take the case of a imiform 
horizontal force 

X «/cos (erf -f- €) (9) 

This will illustrate, to a certain extent, the generation of tides in a land- 
locked sea of small dimensions. 



178-179] 



Waves in a Finite Canal 



257 



AflnnnniTig that ^ varies as cos (<ft + e), and omitting the time-factor, the 
equation (1) becomes 






the solution of which is 



* / , ys . ox , « ox 

t=« — =4+Z)sm — h^cos — . 



(10) 



The terminal conditions give 



al 



<^hf 



p2» 



Z)sin— = (1 — COS — ) ,. 



(11) 



Hence, unless sin a\\c = 0, we have Z) =//<y* . tan (7Z/2c, so that 

2/ 



^ = 



era; . a (Z — a;) , . , . "^ 

sm ;c- sm — h^^ — . cos [pt + €), 



and 



2c 2c 

<7 (a? - \l) 



. cos ((7^ + €). 



(12) 



a* cos (JaZ/c) 

hf 
oc cos ( J a(/c) u 

If the period of the disturbing force be large compared with that of the 
slowest free mode, a{/2c will be small, and the formula for the elevation 
becomes 



=.i(x^ 



(x — ^l) cos (at + €), 



(13) 



approximately, exactly as if the water were devoid of inertia. The horizontal 
displacement of the water is always in the same phase with the force, so long 
as the period is greater than that of the slowest free mode, or al/c < tt. If 
the period be diminished until it is less than the above value, the phase is 
reversed. 

When the period is exactly equal to that of a free mode of odd order 
(« =» 1, 3, 6, . . .), the above expressions for ^ and rj become infinite, and the 
solution fails. As pointed out in Art. 168, the interpretation of this is that, 
in the absence of dissipative forces, the ampUtude of the motion becomes so 
great that our fundamental approximations are no longer justified. 

If, on the other hand, the period coincide with that of a free mode of 
even order (« = 2, 4, 6, . . . ), we have sin al/c = 0, cos al/c = 1, and the terminal 
conditions are satisfied independently of the value of D. The forced motion 
may then be represented by* 



2/ . ax 
= — -_ sin* ^- 



sm* ^ cos (<rf + €). 



(14) 



* In the language of the general theory, the impressed force has here no component of the 
particular type with which it synchronizes, so that a vibration of this tyipe is not excited at aU. 
In the same way a periodic pressure applied at any point of a stretched string will not excite any 
fundamental mode which has a node there, even though it synchronize with it. 

L. H. 17 



258 Tid43d Waves [ohap. vra 

This example illustrates the fact that the effect of a disturbing force may 
often be conveniently calculated without resolving the force into its * normal 
components' (Art. 168). 

Another very simple case of forced oscillations, of some interest in 
connection with tidal theory, is that of a canal closed at one end and 
communicating at the other with an open sea in which a periodic oscillation 

17 = a cos (cri + e) (15) 

is maintained. If the origin be taken at the closed end, the solution is 
obviously 

cos (axle) / . V /-./.v 

{ denoting the length. If aljc be small the tide has sensibly the same 
amplitude at all points of the canal. For particular values of I (determined 
by cos aljc = 0) the solution fails through the ampUtude becoming infinite. 

Canal Theory of the Tides. 

180. The theory of forced osciUations in canals, or on open sheets of 
water, owes most of its interest to its bearing on the phenomena of the tides. 
The * canal theory,' in particular, has been treated very fully by Airy*. We 
will consider one or two of the more interesting problems. 

The calculation of the disturbing effect of a distant body on the waters 
of the ocean is placed for convenience in an Appendix at the end of this 
Chapter. It appears that the disturbing effect of the moon, for example, 
at a point P of the earth's surface, may be represented by. a potential €l 
whose approximate value is 

« = f^(i-C08«^). (1) 

where M denotes the mass of the moon, D its distance from the earth's 
centre, a the earth's radius, y the * constant of gravitation,' and ^ the moon's 
zenith distance at the place P. This gives a horizontal acceleration dn/ad^, 
or 

. /sin2a, .(2) 

towards the point of the earth's surface which is vertically beneath the moon, 
where 






D^ 



(3) 



* EjicgcL Mdrop, *' Tides and Waves," Section vi. (1845). Several of the leading features of 
the theory had been made out, by very simple methods, by Young, in 1813 and 1823 [Works, t. ii. 
pp. 262. 291]. 



179-181] Canal Theory of the Tides 259 

If E be the earth's mass, we may write g » yE/a^, whence 

/ 3 M /a\8 
g^2' E '[dJ ' 

Putting M/E = ^, a/Z) = ^, this gives f/g = 8-57 x lO-®. When the sun is 
the disturbing body, the corresponding ratio iaf/g = 3*78 x 10~®. 

It is convenient, for some purposes, to introduce a Unear magnitude £r, 
defined by 

H'-afig (4) 

If we put a == 21 X 10* feet, this gives, for the lunar tide, H = 1*80 ft., and 
for the solar tide H = '79 ft. It is shewn in the Appendix that H measures 
the maximum range of the tide, from high water to low water, on the * equi- 
librium theory.' 

181. Take now the case of a imiform canal coincident with the earth's 
equator, and let us suppose for simplicity that the moon describes a circular 
orbit in the same plane. Let ^ be the displacement, relative to the 
earth's surface, of a particle of water whose mean position is in longitude 
</}, measured eastwards from some fixed meridian. If co be the angular 
velocity of the earth's rotation, the actual displacement of the particle at 
time t will be f + «a>f, so that the tangential acceleration will be d^^/dfi. 
If we suppose the 'centrifugal force' to be as usual allowed for in the value 
of g, the processes of Arts. 169, 177 will apply without further alteration. 

If n denote the angular velocity of the moon westward, relative to the fixed 
meridian*, we may write in Art. 180 (2) 

a^ = w« + ^ + €, 

so that the equation of motion is 

P = ''*a-|f«--^«^2(««+^ + e) (1) 

The free oscillations are determined by the consideration that ^ is 
necessarily a periodic function of ^, its value recurring whenever <f> increases 
by 2n. It may therefore be expressed, by Fourier's Theorem, in the form 

I = S (P^cosr^ + g^sinr<^) (2) 



Substituting in (1), with the last term omitted, it i^' found that P^ and Q^ 
must satisfy the equation 

dip «.2^2 

The motion, in any normal mode, is therefore simple-harmonic, of period 
27ralrc. 

* That is, nsw-Tii^ifn^be the angular velocity of the moon in her orbit. 

17—2 



260 Tidal Waves [ohap. vin 

For the forced waves, or tides, we find 

^--i^^^2^^^(^ + <f>-^^)^ W 

whence iy = ^ c« - n^a^ cos 2 (nt + ^ + c) (5) 

The tide is therefore semi-diurnal (the lunar day being of course understood), 
and is 'direct' or 'inverted,' i.e. there is high or low water beneath the moon, 
according as c ^ na, in other words according as the velocity, relative to the 
earth's surface, of a point which moves so as to be always vertically beneath 
the moon, is less or greater than that of a free wave. In the actual case of 
the earth we have 



n^a^ n^a a a 

so that unless the depth of the canal were to greatly exceed such depths as 
actually occur in the ocean, the tides would be inverted. 

This residt, which is sometimes felt as a paradox, comes under a general 
principle referred to in Art. 168. It is a consequence of the comparative 
slowness of the free (i^illations in an equatorial canal of moderate depth. 
It appears from the rough numerical table on p. 249 that with a depth 
of 11250 feet a free wave would take about 30 hours to describe the earth's 
semi-circumference, whereas the period of the tidal disturbing force is only a 
little over 12 hours. 

The formida (5) is, in fact, a particular case of Art. 168 (14), for it may 
be written 

where rj is the elevation given by the 'equilibrium theory,' viz. 

^ = Jff cos 2 (n^ -f ^ -f- €), (7) 

and a =» 2n, Gq = 2c/a. 

For such moderate depths as 10000 feet and under, n^a^ is large com- 
pared with gh ; the amphtude of the horizontal motion, as given by (4), is 
then //4n2, or gj^rt^a . £f, nearly, being approximately independent of the 
depth. In the case of the limar tide this amplitude is about 140 feet. The 
maximum elevation is obtained by multiplying by 2%/a; this gives, for a 
depth of 10000 feet, a height of only 133 of a foot. 

For greater depths the tides would be higher, but still inverted, until 
we reach the critical depth n^a^/g, which is about 13 miles. For depths 
beyond this limit, the tides become direct, and approximate more and more 
to the value given by the equilibrium theory*. 

♦ Cf. Young, Lc. ante p. 2C8. 



181-183] Tide in Equatorial Canal 261 

182. The case of a circular canal parallel to the equator can be worked 
out in a similar manner. If the moon's orbit be still supposed to he in the 
plane of the equator, we find by spherical trigonometry 

cos ^ = sin cos (n^ + <^ + €), (1) 

where is the co-latitude, and <f> the longitude. The disturbing force in 
longitude is therefore 

^-^^ = -/sin flsin 2{nt + d>-{-€) (2) 

Thisleadsto ^ = J ^^-^^^1^^^ (3) 

Hence if wa > c the tide will be direct or inverted according as 5 $ sin"* c\na. 
If the depth be so great that ona^ the tides will be direct for all values 
of 0. 

If the moon be not in the plane of the equator, but have a co-declination 
A, the formula (1) is replaced by 

cos & = cos ^ cos A + sin sin A cos a, (4) 

where a is the hour-angle of the moon from the meridian of P. For 
simplicity, we will neglect the moon's motion in declination in comparison 
with the earth's angular velocity of rotation ; thus we put 

a » «i + ^ + €, 

and treat A as constant. The resulting expression for the disturbing force 
along the parallel is found to be 

oiii = —/cos 6 sin 2A sin (n* + ^ + €) 

-/sin Q sin« A sin 2 (n< + ^ + «) (5) 

We thence obtain 

17 = i -^ ^ , . o -qSU^ 20 sin 2A cos (n^ + <A + c) 

+ \ 1 , o . .^ sin* e sin« A cos 2 (n^ + <^ + €) (6) 

The first term gives a 'diurnal' tide of period %T\n\ this vanishes and 
changes sign when the moon crosses the equator, i,e, twice a month. The 
second term represents a semi-diurnal tide of period 7r/n, whose ampUtude is 
now less than before in the ratio of sin* A to 1. 

183. In the case of a canal coincident with a meridian we should have 
to take account of the fact that the undisturbed figure of the free surface 
is one of relative equilibrium under gravity and centrifugal force, and is 
therefore not exactly circular. We shall have occasion later on to treat the 
question of displacements relative to a rotating globe somewhat carefully; 
for the present we will assume by anticipation that in a narrow canal the 



262 Tidal Waves [chap, vm 

disturbances are sensibly the same as if the earth were at rest, and the 
disturbing body were to revolve round it with the proper relative motion. 

If the moon be supposed to move in the plane of the equator, the hour- 
angle from the meridian of the canal may be denoted by w^ + €, and if d be 
the co-latitude of any point P on the canal, we find 

cos Sr = sin . cos (n^ + c) (1) 

The equation of motion is therefore 

Solving, we find 

T? = - iff cos 2^ - i -^ 2—2 ^^8 25 • ^^ 2 (n« + €) (3) 

The first term represents a permanent change of mean level to the extent 

7^ = - Jffcos2e (4) 

The fluctuations above and below the disturbed mean level are given by 
the second term in (3). This represents a semi-diurnal tide ; and we notice 
that if, as in the actual case of the earth, c be less than na, there will be 
high water in latitudes above 45°, and low water in latitudes below 45°, when 
the moon is in the meridian of the canal, and vice versa when the moon is 
90° from that meridian. These circumstances would be all reversed if c were 
greater than na. 

When the moon is not on the equator, but has a given declination, the 
mean level, as indicated by the term corresponding to (4), has a coefficient 
depending on the declination, and the consequent variations in it indicate a 
fortnightly (or, in the case of the sun, a semi-annual) tide. There is also 
introduced a diurnal tide whose sign depends on the declination. The reader 
will have no difficulty in examining these points, by means of the gene];al 
value of ii given in the Appendix. 

• 

184. In the case of a uniform canal encircling the globe (Arts. 181, 182) 
there is necessarily everywhere exact agreement (or exact opposition) of phase 
between the tidal elevation and the forces which generate it. This no longer 
holds, however, in the case of a canal or ocean of limited extent. 

Let us take for instance the case of an equatorial canal of finite length. 
If the origin of time be suitably chosen we have 

3=''*^*--^«^'^2(«^+^). (1) 

with the condition that £ = at the ends, where ^ = ± a, say. 



183-184] Tid^ in Finite Canal 263 

If we neglect the inertia of the water the term d^i/dfi is to be omitted, 
and we find 

f = J'^-]8in2w^cos2a + -cos2w^sin2a — sin 2 (n< + ^)[. ..(2) 

Hence 17 = - *|| = Ji? |cos2 (n^ + <^) - ?^cos2n«l, (3) 

where H ^^fa/g, as in Art. 180. This is the elevation on the (corrected) 
'equiUbrium' theory referred to in the Appendix to this Chapter. At the 
centre {<f> = 0) of the canal we have 

,, = Jffcofl2««(l-^) (4) 

If a be small the range is here very small, but there is not a node in the absolute 
sense of the term. The times of high water coincide with the transits of 
moon and * anti-moon.'* At the ends ^ = ± a we have 

71 = iff |(l -^-^^) cos2 (n<± a)T ^'^^^ Bm2{nt±a) 



= iHRo cos 2 (n« ± a =F cq), (5) 

sin 4a » • « 1 — cos 4a 
4^' /?o8m2eo= ^ 



if Rq cos 2€o = 1 z — , Rq sm 2^0 = r (^) 



Here €0 denotes the hour-angle of the moon W. of the meridian when 
there is high water at the eastern end of the canal, or E. of the meridian 
when there is Idgh water at the western end. When a is small we have 

Bo = 2a, €o=-iir + |a, (7) 

approximately. 

When the inertia of the water is taken into account we have 

f = L o IV a I sin 2 {nt + ^) - -=-4 — {sin 2(fU + a) sin 2m {<t> + a) 

— sin 2 (w< — a) sin 2m (^ — a)} , (8) 

where m = na/c. Hence f 

■n = — 1 -X r COS 2int-\-6) r—T- {sin 2 (n^ + a) cos 2m (tk + a) 

' ^ m* — 1 L sm 4ma ^^ . ' 



— sin 2 (w< — a) cos 2m (^ — a)} (9) 



If we imagine w to tend to the limit we obtain the formula (3) of the 
equilibrium theory. It may be noticed that the expressions do not become 

* This term is explained in the Appendix to this Chapter. 

t Of. Airy, "Tides and Waves," Art. 301. The discussion in the text is from a paper in 
the Phil Mag, (6), t. xxix. p. 737 (1915). 



264 



Tidal Waves 



[chap, vm 



infinite f or m -^ 1 as they do in the case of an endless canal. In all cases 
which are at all comparable with oceanic conditions m is, however, considerably 
greater than unity. 

At the centre of the canal we have 

H o ^ /n m sin 2a\ 

cos 2nt 1 1 ; — s — 

\ sin2ma/ 



,=-j 



(10) 



m* — 1 \ sm 

As in the eqniUbrium theory, the range is very smaU if a be small, but there 
is not a true node. At the ends we find 



H 



"^^^W^^ 



m sin 4a 
\ sin 4ma 



— ljcos2 (vJt± a) 



m (cos ima — cos 4a) . « , ^ . x 
sm 4ma 



if 

Bi cos 2€i = 



(11) 



= \HR^ cos 2 (n^ ± a qpci), 

m sin 4a — sin 4ma ^ . ^ m (cos 4ma — cos 4a) .-^v 

, Ri sm 2€i = -7—= — -. . . . . .(12) 



(m* — 1) sm 4ma 



(13) 



(m* — 1) sin 4ma 
When a \a small we have 

Ri = 2a, €1 = — Jtt + fa, 
approximately, as in the case of the equilibrium theory. 

The value of Ri becomes infinite when sin 4ma » 0. This determines the 
critical lengths of the canal for which there is a free period equal to 7r/n, 
or half a lunar day. The limiting value of c^ in such a case is given by 

tan 2€i = — cot 2a, or = tan 2a, 

according as 4ma \s an odd or even multiple of tt. 



• 


Corrected Equilibrium Theory 


Dynamical Theory 


2a 


2aa 


Range at 


Range at 


fo 


Range at 


Range at 


1 
€1 


(degrees) 


(miles) 


centre 


ends 


(degrees) 


centre 


ends 


(degrees) 














-45 








-45 


9 


540 


•004 


•167 


-42 


•004 


•166 


-41-9 


18 


1080 


•016 


•311 


-39 


•018 


•396 


-38-5 


27 


1620 


•037 


•460 


-36 


•044 


•941 


-33-9 


31-5 


1890 


•050 


•531 


-34-5 


•063 


1946 


-30-9 


36 


2160 


•065 


•601 


-33 


•089 


00 


(-27 
\+63 
+ 68-2 


40-5 


2430 


•081 


668 


-31-6 


•125 


1-956 


45 


2700 


•100 


•733 


-301 


•174 


•987 


+ 75-7 


54 


3240 


•142 


•853 


-27-2 


•354 


•660 


-83-5 


63 


3780 


•190 


•969 


-24 4 


•918 


1141 


-65-1 


72 


4320 


•243 


1051 


-216 


00 


00 


j -54 
\+36 
+ 44 5 


81 


4860 


•301 


M27 


-18-9 


h459 


1112 


90 


6400 


•363 


M85 


-162 


•864 


•613 


+ 55-9 



184-rl85] Canal of Varying Section 265 

The table illustratee the ease of m =2*5. If irjn — 12 lunar hours this implies a depth 
of 10820 ft., which is of the same order of magnitude as the mean depth of the ocean. 
The corresponding wave-velooity is about 360 sea-miles per hour. The first critical 
length is 2160 miles (a =t\r^)- ^^® ^^^ ^ terms of which the range is expressed is the 
quantity H, whose value for the lunar tide is about 1*80 ft. The hour-angles cq and f^ 
are adjusted so as to lie always between ±: 90°, and the positive sign indicates position W. 
of the meridian in the case of the eastern end of the canal, and E. of the meridian for the 
western end. 

Wave-Motion in a Canal of Variable Section, 

185. When the section (S, say) of the canal is not unifonn, but varies 
gradually from point to point, the equation of continuity is, by Art. 
169 (11), 

where b denotes the breadth at the surface. If h denote the mean depth 
over the width .6, we have S = bh, and therefore 

^--lrx<'^^^' (2) 

where h, b are now functions of z. 

The dynamical equation has the same form as before, viz. 

w=-nx (^) 

Between (2) and (3) we may eliminate either ry or f ; the equation in 17 is 



^=\U«'l) <*) 



The laws of propagation of waves in a canal of gradually varying rect- 
angular section were investigated by Green*. His results, freed from the 
restriction to the special form of section, may be obtained as follows. 

If we introduce a variable r defined by 

Tr-(9^^ W 

in place of x, the equation (4) transforms into 

where the accents denote differentiations with respect to t. If 6 and h were constcuits, the 
equation would be satisfied hy rj = F {t - 1), aa ia. Art. 170 ; in the present case we assume, 
for trial, 

Ti=e.F(T-t), (7) 

where 6 is a function of r only. Substituting in (6), we find 



^e' F' e" fh' \hr\(F' e'\ ^ ,«. 



• "On the Motion of Waves in a Variable Canal of small depth and width," Camb. Trans. 
t. vi. (1837) [Papers, p. 225]; see also Airy, "Tides and Waves," Art. 2«0. 



266 Tidai Waves [chap, vin 

The terms of this which involve F will cancel provided 

or e = 06 " i A ~ i ( 9 ) 

C being a constant. Hence, provided the remaining terms in (8) may be neglected, the 
equation (4) will be satisfied. 

The above approximation is justified, provided we can neglect 0"lB' and 676 in com- 
parison with F'/F. As regards 676, it appears from (9) and (7) that this is equivalent to 
neglecting b~^ . db/dx and h~^ . c^fdx in comparison with rj"^ . drj/dz. If, now, X denote a 
wave-length, in the general sense of Art. 172, dri/dx is of the order 17/X, so that the assump- 
tion in question is that Tidb/dx and ^dh/dx are small compared with b and h, respectively. 
In other words, it is assumed that the transverse dimensions of the canal vary only by 
small fractions of themselves within the limits of a wave-length. It is easily seen, in like 
manner, that the neglect of 6^7^' ^ comparison with F'/F implies a similar limitation to 
the rates of change of db/dx and dh/dx. 

Since the equation (4) is unaltered when we reverse the sign of t, the complete solution, 
subject to the above restrictions, is 

,; =6 -4 A-i {^ (r - +/(r +t)} (10) 

where F and / are arbitrary functions. 

The first term in this represents a wave travelling in the direction of x-positive ; the 
velocity of propagation past any point is determined by the consideration that any particular 
phase is recovered when br and lit have equal values, and is therefore equal to *J{gk), by 
(5), as we should expect from the case of a uniform section. In like manner the second 
term in (10) represents a wave travelling in the direction of ar-negative. In each case the 
elevation of any particular part of the wave alters, as it proceeds, according to the law 

b-h-i. 

The reflection of a progressive wave at a point where the section of a 
canal suddenly changes has been considered in Art. 176. The formulae there 
given shew, as we should expect, that the smaller the change in the 
dimensions of the section, the smaller will be the amplitude of the reflected 
wave. The case where the change from one section to the other is 
continuous, instead of abrupt, has been investigated by Kayleigh for a 
special law of transition*. It appears that if the space within which the 
transition is completed be a moderate multiple of a wave-length there is 
practically no reflection; whilst in the opposite extreme the results agree 
with those of Art. 176. 

If we assume, on the ba.sis of these results, that when the change of 
section within a wave-length may be neglected a progressive wave suffers 
no appreciable disintegration by reflection, the law of amplitude easily follows 
from the principle of energy f. It appears from Art. 174 that the energy of 

* "On Reflection of Vibrations at the Confines of two Media between which the Transition is 
gradual," Proc, Lond, Math, Soc. t. zi. p. 51 (1880) [Papers, t. i. p. 460]; Theory of Sound, 2nd^., 
London, 1894, Art 1486. 

t Rayleigh, Ic. anU p. 262. 



186-186] Canal of Varying Section 267 

the wave varies as the length, the breadth, and the square of the height, and 
it is easily seen that the length of the wave, in different parts of the canal, 
varies as the corresponding velocity of propagation, and therefore as the square 

root of the mean depth. Hence, in the above notation, ifbhr is constant, or 

which is Green's law above found. 

186. In the case of simple harmonic motion, where t] oc cos (at + e). the 
equation (4) of the preceding Art. becomes 

f5(»»l)+-'-« m 

Some particular cases of considerable interest can be solved with ease. 

1^. For example, let us take the case of a canal whose breadth varies as 

the distance from the end a; = 0, the depth being uniform ; and let us suppose 

that at its mouth (x = a) the canal communicates with an open sea in which 

a tidal oscillation 

7^ = C cos ((7« + €) (2) 

is maintained. Putting h = const., 6 oc x, in (1), we find 

D+il+'''-» '" 

provided A;* = o^jgh (4) 

Hence -n = C / L , cos (a< + c) (5) 

Jo(*a) 

The curve y ^^ Jq (x) is figured on p. 278 ; it indicates how the amplitude 

of the forced oscillation increases, whilst the wave-length is practically 

constant, a.s we proceed up the canal from the mouth. 

2^. Let us suppose that the variation is in the depth only, and that this 
increases uniformly from the end x = of the canal to the mouth, the remain- 
ing circumstances being as before. If, in (1), we put h = hQx/a, k = ci^ajgh^y 
we obtain 

i('i)+''-» <" 

The solution of this which is finite for x = is 

. /- KX K^X^ \ .„. 

r) = All — Y^ -{- ^2 22 "~ • • • ] » I ' / 

or 7] = AJo (2/c*x*), (8) 

whence finally, restoring the time-factor and determining the constant, 

= 0«M?!L?) cos (<H + €) (9) 



268 



Tidal Waves 



[chap, vin 



The annexed diagram of the curve y = Jq (V^), where, for clearness, the 
scale adopted for y is 200 times that of x, shews how the amplitude continually 
increases, and the wave-length diminishes, as we travel up the canal. 

These examples may serve to illustrate the exaggeration of oceanic tides 
which takes place in shallow seas and in estuaries. 




We add one or two simple problems of free oscillations. 

3^. Let us take the case of a canal of uniform breadth, of length 2a, whose 
bed slopes uniformly from either end to the middle. If we take the origin at 
one end, the motion in the first half of the canal will be determined, as 
above, by 

7? = 4 Jo (2#cM), (10) 

where k ^ o^djgh^^ h^ denoting the depth at the middle. 

It is evident that the normal modes will fall into two classes. In the first 
of these 77 will have opposite values at corresponding points of the two halves 
of the canal, and will therefore vanish at the centre (x = a). The values of a 
are then determined by 

Jo(2A*) = 0, (11) 

viz. #c being any root of this, we have 

„^i9j^,^^a)i (12) 

In the second class, the value of t) is symmetrical with respect to the 
centre, so that dri/dx = at the middle. This gives 

Jo'(2A*) = (13) 

It appears that the slowest oscillation is of the asymmetrical class, and 
corresponds to the smallest root of (11), which is 2ic*a* = -TeSSw, whence 

^ . 1-306 X -*^. 



186-187] Canal of Varying Section 269 

4^. Again, let us suppose that the depth of the canal varies according to 
the law 

A = Ao(l-^), (14) 

where x now denotes the distance from the middle. Substituting in (1), with 
6 = const., we find 

4{('-s:)l}-£'-» <>») 

If we put ff» = n (n + 1)^, (16) 

this is of the same form a.s the general equation of zonal harmonics, Art. 
84 (1). 

In the present problem n is determined by the condition that 77 must be 
finite for xja = ± 1. This requires (Art. 85) that n should be integral; the 
normal modes are therefore of the type 



= CP, (?) . cos M -h €), (17) 



where P„ is a zonal harmonic, the value of a being determined by (16). 

In the slowest oscillation (n = 1), the profile of the free surface is a 
straight line. For a canal of uniform depth h^, and of the same length (2a), 
the corresponding value of a would be 7rc/2a, where c = {gh^ , Hence in the 
present case the frequency is less, in the ratio 2\/27r, or -9003*. 

The forced oscillations due to a uniform disturbing force 

X =/cos ((7« + e) (18) 

can be obtained by the rule of Art. 168 (14). The equilibrium form of the 
free surface is evidently 

^ =-^ X cos (a« -h €), (19) 

and, since the given force is of the normal type w = 1, we have 

^ = (y (1 -Vo') '^ '^"^ <°* + '^' (20) 

where ag* = ^gh^ja!^. 

Waves of Finite Amplitude. 

187. When the elevation i) is not small compared with the mean depth 
A, waves, even in an uniform canal of rectangular section, are no longer 
propagated without change of type. The question was first investigated by 

* For extenBions, and applications to the theory of * seiches* in lochs, see Chrystal, "Some 
Results in the Mathematical Theory of Seiches," Proc. R. S. Edin. t. xxv. p. 328 (1904), and 
Tran9, R, 8, Edin, t. xli. p. 699 (1906). 



I 

I 



270 Tidal Waves [ohap. vra 

Airy*, by methods of successive approximation. He found that in a pro* 
gressive wave different parts will travel with different velocities, the wave- 
velocity corresponding to an elevation 77 being given approximately by 

»(1H-Ii), 

where c is the velocity corresponding to infinitely small amplitude. 

A more complete view of the matter can be obtained by the method 
employed by Biemann in treating the analogous problem in Acoustics, to 
which reference will be made in Chapter x. 

The sole assumption on which we are now proceeding is that the vertical 
acceleration may be neglected. It follows, as explained in Art. 168, that 
the horizontal velocity may be taken to be uniform over any section of the 
canal. The djmamical equation is 

du , du dri 

¥ + «^=-^ai' (1) 

as before, and the equation of continuity, in the case of a rectangular section, 
is easily seen to be 

a^P + '?)«}=-!. ; (2) 

where h is the depth. This may be written 

| + «^--(» + ,)g (3) 

Let us now write 

P =/(!,) + M, Q =/(,,) -u, (4) 

where the function / (rj) is as yet at our disposal. If we multiply (3) by 
/' (tj), and add to (1), we get 

If we now determine/ (tj) so that 

{h + v){f'm* = 9 (5) 

this may be written 

¥ + «I=-(* + ''>/'('')S («) 

In the same way we find 

f + .g- (»+,)/'wg m 

The condition (5) is satisfied by 

/(,) = 2c|(l + |)*-lj. (8) 

♦ "Tides and Waves," Art. 198. 



187] Waves of Finite Amplitude 271 

where c^ {g^) - Tbe arbitrary constant has been chosen so as to make 
P and Q vanish in the parts of the canal which are free from disturbance, 
but this is not essential. 

Substituting in (6) and (7) we find 

tf-[&-{.(.+i/+„}*]g,^ 

V . ^«^^ 

*.[&+jc(l+|)»-„}*]|. 

It appears, therefore, that dP = 0, i,e, P is constant, for a geometrical point 
moving with the velocity 

S-('^l)*+». w 

whilst Q is constant for a point moving with the velocity 

i--«('-^i)'^« <"' 

Hence any given value of P travels forwards, and any given value of Q travels 
backwards, with the velocities given by (10) and (11) respectively. The 
values of P and Q are determined by those of i] and w, and conversely. 

As an example, let us suppose that the initial disturbance is confined 
to the space for which a<x<h, so that P and Q are initially zero for 
X <a and x>h. The region within which P differs from zero therefore 
advances, whilst that within which Q differs from zero recedes, so that after 
a time these regions separate, and leave between them a space within which 
P = 0, = 0, and the fluid is therefore at rest. The original disturbance 
has now been resolved into two progressive waves travelling in opposite 
directions. 

In the advancing wave we have 

= 0, iP = i* = 2c{(l+|)*-lj,.... (12) 

so that the elevation and the particle- velocity are connected by a definite 
relation (cf. Art. 171). The wave-velocity is given by (10) and (12), viz. it is» 



c 



H'+D'-^l <") 



To the first order of t^/A, this is in agreement with Airy's result. 

Similar conclusions can be drawn in regard to the receding wave*. 

Since the wave- velocity increases with the elevation, it appears that in 
a progressive wave-system the slopes will become continually steeper in front, 
and more gradual behind, until at length a state of things is reached in 

* The above reeulta can also be deduced from the equation (3) of Art. 173, to which RiemaDD*8 
method can readily be adapted. 



272 Tidal Waves [chap, vni 

which we are no longer justified in neglecting the vertical acceleration. As 
to what happens after this point we have at present no guide from theory; 
observation shews, however, that the crests tend ultimately to curl over and 
break. 

The case of a *bore,' where there is a transition from one uniform level 
to another, may be investigated by the artifice of steady motion (Art. 175). 
If Q denote the volume per unit breadth which crosses each section in unit 
time we have 

Uyh^ = u^h^ = e, (14) 

where the sufiGlxes refer to the two uniform states, h^ and A^ denoting the 
depths. Considering the mass of fluid which is at a given instant contained 
between two cross-sections, one on each side of the transition, we see that in 
unit time it gains momentum to the amount pQ (u^ — i^), the second section 
being supposed to lie to the right of the first. Since the mean pressures over 
the sections are ^gphi and ^gph^, we have 

Q {«. - «i) = isr (^ - A,) (15) 

Hence, and from (14), 

Q» = yh^ht ih + ht) (16) 

If we impress on everything a velocity — Mj we get the case of a wave invading 
still water with a velocity of propagation 



«! = V^f 



2., J (") 



in the negative direction. The particle-velocity in the advancing wave is 
Wj — Mj in the direction of propagation. This is positive or negative according 
as ^2 < ^1) ^•^* according as the wave is one of elevation or depression. 

The equation of energy is however violated, imless the difference of level be 
regarded as infinitesimal. If, in the steady motion, we consider a particle 
moving along the surface stream-line, its loss of energy in passing the place 
of transition is 

ip W - V) -\-9Pih-K) (18) 

per unit volume. In virtue of (14) and (16) this takes the form 

gp (h^ - Ai)« 

4A1A3 ^^^^ 

Hence, so far as this investigation goes, a bore of elevation {h^ > hi) can 
be propagated unchanged on the assumption that dissipation of energy takes 
place to a suitable extent at the transition. If however ^2 < ^1 > ^^^ expression. 
(19) is negative, and a supply of energy would be necessary. It follows that 
a negative bore of finite height cannot in any case travel unchanged*. 

♦ Rayleigh, " On the Theory of Long Waves and Bores," Proc, Roy, Soc. A, t. xo. p. 324 (1914). 



187-188] Tides of Second Order 273 

188. In the detailed application of the equations (1) and (3) to tidal 
phenomena, it is usual to follow the method of successive approximation. 
As an example, we will take the case of a canal communicating at one end 
{x = 0) with an open sea, where the elevation is given by 

7^ = a cos at (20) 

For a first approximation we have 

¥-"^S' ¥-"^^ ^^^' 

the solution of whioh, consistent with (20), is 

rf=a COR alt J, u=: — 0OB<r(t ) (22) 

For a second approximation we substitute these values of rj and i^ in (1) and (3), and obtain 

¥ = -^S-V'^2<r^t--j. ^ = _A^-^Bm2<r(«--j. ...(23) 
Integrating these by the usual methods, we find, as the solution consistent with (20), 

y • • • • I 



get 

U = ^— cos cr 
c 



(..?)-i^%os2.(.-f)-i?!^%sin2.(.^?).J 



(24) 



The annexed figure shews, with, of course, exaggerated amplitude, the profile of the 
waves in a particular case, as determined by the first of these equations. It is to be noted 
that if we fix our attention on a particular point of the canal, the rise and fall of the 
water do not take place symmetrically, the fall occupying a longer time than the rise. 



The occurrence of the factor x outside trigonometrical terms in (24) shews that there is 
a Hmit beyond which the approximation breaks down. The condition for the success of 
the approximation is evidently that gtrax/c^ should be small. Putting c*=gh, X=29rc/o', 
this fraction becomes equal to 2ir (a/h) . (x/X). Hence however small the ratio of the 
original elevation (a) to the depth, the fraction ceases to be small when x is a sufficient 
multiple of the wave-length (X). 

It is to be noticed that the Hmit here indicated is already being overstepped in the 
right-hand portions of the figure; and that the peculiar features which are beginning 
to shew themselves on the rear slope are an indication rather of the imperfections 
of the analysis than of any actual property of the waves. If we were to trace the 
curve further, we should find a secondary maximum and minimum of elevation developing 
themselves on the rear slope. In this way Airy attempted to explain the phenomenon 
of a double high-water which is observed in some rivers; but, for the reason given, the 
argument cannot be sustained *. 

The same difficulty does not necessarily present itself in the case of a canal closed by a 
fixed barrier at a distance from the mouth, or, again, in the case of the forced waves due to 

* MoCowan, Ix. ante p. 251. 
L. H. .18 



274 Tided Waves [chap, vra 

a periodio horizontal force in a canal dosed at both ends (Art. 179). Enough has, however, 
been giv^ to shew the general character of the results to be expected in such cases. For 
further details we must refer to Airy*s treatise*. 

When analysed, as jn (24), into a series of simple-harmonic functions of the time, the 
expression for the elevation of the water at any particular place (x) consists of two terms, 
of which the second represents an 'over- tide,' or 'tide of the second order,' being propoiv 
tional to a'; its frequency is double that of the primary disturbance (20). If we were to 
continue the approximation we should obtain tides of higher orders, whose frequencies are 
3, 4, . . . times that of the primary. 

If, in place of (20), the disturbance at the mouth of the canal were given by 

f =a cos ai'\-a' cos (o-'t + c), 

it is easily seen that in the second approximation we should in like manner obtain tides of 
periods 2w/(o- + a) and 27r/(o' — cr') ; these are called 'compound tides.' They are analogous 
to the 'combination- tones' in Acoustics which were first investigated by Helmholtzf. 

Propagation in Two Dimensions. 

189. Let us suppose, in the first instance, that we have a plane sheet of 
water of uniform depth h. If the vertical acceleration be neglected, the 
horizontal motion will as before be the same for all particles in the same 
vertical line. The axes of x, y being horizontal, let w, v be the component 
horizontal velocities at the point (x, y), and let ^ be the corresponding 
elevation of the free surface above the undisturbed level. The equation of 
continuity may be obtained by calculating the flip: of matter into the 
columnar space which stands on the elementary rectangle hxhy ; viz. we have, 
neglecting terms of the second order, 

^ (uhhy) 8x + U^hhx) 8y = - I {(J + h) 8x8y}, 

^^«°^« i = -Hl+|) (1) 

The dynamical equations are, in the absence of disturbing forces, 

du _^ dp 3v _ dp 

where we may write 

if Zo denote the ordinate of the free surface in the undisturbed state. We 
thus obtain 

du _ 9 J 3v _ 3J 

¥"*"**ai' a?"""^a^ ^^^ 

♦ "Tides and Waves," Arts. 198, ... and 308. See also G. H. Darwin, "Tides," Encyc. 
Britann, (9th ed.) t. xziii. pp. 362, 363 (1888). 

t "Ueber Combinationstone," Berl Monatsber, May 22, 1866 [Wiss. Ahh. t. i. p. 266]; and 
"Theorie der Luftsohwingungen in Rohren mit offenen Enden," CreUe, t. Ivii. p. 14 (1869) 
[Wiss. Abh. t. i. p. 318]. 



188-189] Waves on an Open Sheet of Water 276 

If we eUminate u and v, we find 

where c^ = gh ba before. 

In the application to simple-hannonic motion, the equations are shortened 
if we assume a complex time-factor «* <'*+•', and reject, in the end, the 
imaginary parts of our expressions. This is legitimate so long as we have 
to deal solely with linear equations. We have then, from (2), 

« = ^P. « = *^|-^ (4) 

a ox' a dy ^ ' 

whilst (3) becomes 

3^ + 3^ + ^^"^' ^^) 

where h^ = a^/c^ (6) 

The condition to be satisfied at a vertical bounding wall is obtained at 
once from (4), viz. it is 

i-». <') 

if Sn denote an element of the normal to the boundary. 

When the fluid is subject to small disturbing forces whose variation 
within the limits of the depth may be neglected, the equations (2) are 
replaced by 

at"" ^dx dx' dt~ ^dy dy' ^^ 

where CI is the potential of these forces. 

If we put f = - Q/^, (9) 

80 that ^ denotes the equilibrium-elevation corresponding to the potential Q, 
these may be written 

s-4«-a s-4«-fi "») 

In the case of simple-harmonic motion, these take the forms 

"-!s«-o. "-H«-" <"» 

whence, substituting in the equation of continuity (1), we obtain 

( V + *«) C = V,af, (12) 

^ ^^*=^ + aT- (1') 

18—2 



276 Tidal Waves [ohap. vm 

and 1^ = a^lgh, ba before. The condition to be satisfied at a vertical boundary 
is now 

a|(^-?) = o (1*) 

190. The equation (3) of Art. 189 is identical in form with that which 
presents itself in the theory of the transverse vibrations of a uniformly 
stretched membrane. A still closer analogy, when regard -is had to the 
boundary-conditions, is furnished by the theory of cylindrical waves of 
sound*. Indeed many of the results obtained in this latter theory can be 
at once transferred to our present subject. 

Thus, to find the free oscillations of a sheet of water bounded by vertical 
walls, we require a solution of 

(V,« + A»)S = (1) 

subject to the boundary-condition 

g-» p) 

Just as in Art. 178 it will be found that such a solution is possible only for 
certain values of k, which accordingly determine the periods (^n/kc) of the 
various normal modes. 

Thus, in the case of a rectangular boundary, if we take the origin at one 
corner, and the axes of x, y along two of the sides, the boundary-conditions 
are that dt^jdx = for x = and x = a, and dl,jdy = f or y = and y = b, 
where a, 6 are the lengths of the edges parallel to x, y respectively. The 
general value of J subject to these conditions is given by the double Fourier's 
series 

i = SSil„,„ COS -^ cos -^, (3) 

where the summations include all integral values of m, n from to oo » 
Substituting in (1) we find 






If a > 6, the component oscillation of longest period is got by making w = 1, 
n = 0, whence ka = n. The motion is then everywhere parallel to the longer 
side of the rectangle. Cf. Art. 178. 

191. In the case of a circular sheet of water, it is convenient to take 
the origin at the centre, and to transform to polar co-ordinates, writing 

x = r cos 0y y = rsiad. 

♦ Rayleigh, Theory of Sound, Art. 339. 



189-191] Circular Basin 277 

The equation (1) of the preceding Art. becomes 

dr^^rd^^V^W'^''^ ^ (^) 

This might of course have been established independently. 

As regards dependence on 0, the value of J may, by Fourier's Theorem, 
be supposed expanded in a series of cosines and sines of multiples of ; we 
thus obtain a series of terms of the form 

/(^)IV <') 

It is found on substitution in (1) that each of these terms must satisfy the 
equation independently, and that 



rir) + lf'ir) + (h*-^f{r)^0 (3) 



This is of the same form as Art. 101 (14). Since f must be finite for 
r = 0, the various normal modes are given by 



cos"" 



J = A,J, (hr) . y8d,coB{at + €), (4) 

where 8 may have any of the values 0, 1, 2, 3, ... , and A, is an arbitrary 
constant. The admissible values of k are determined by the condition that 
d^/dr = at the boundary r = a, say, or 

J/{ka)=^0 (5) 

The corresponding 'speeds' (a) of the oscillations are then given by 

In the ca.se 8 — 0, the motion is symmetrical about the origin, so that the 
waves have annular ridges and furrows. The lowest roots of 

Jo' (*a) = 0, or Ji (*a) = 0, (6) 

are given by 

Aw/tt = 1-2197, 2-2330, 3-2383, . . . , (7) 

these values tending ultimately to the form ka/'jr = m-\- 1, where m is 
integral*. In the mth mode of the symmetrical class there are m nodal 
circles whose radii are given by J = or 

Jo(*^) = (8) 

The roots of thisf are 

jfcr/TT = -7656, 1-7571, 2-7546, (9) 

* Stokes, **0n the Numerioal Galoulation of a class of Definite Integrals and Infinite Series," 
Comb. Trans, t. ix. (1850) [Papere, t. ii. p. 355]. 

It is to be noticed that kajv is equal to tJt, where r is the actual period, and Tq is the time a 
progressive wave would take to travel with the velocity *J{gh) over a space equal to the 
diameter 2a. 

t Stokes, ^e. 



278 



Tidal Waves 



[chap, vra 




I 



o 

Ik 

S 



JS 






191] Circular Basin 279 

For example, in the first symmetrical mode there is one nodal circle r = •628a. 
The form of the section of the free surface by a plane through the axis of 2, 
in any of these modes, will be understood from the drawing of the curve 
y = •'^0 (^)> which is given on the preceding page. 

When 8>Q there are 8 equidistant nodal diameters, in addition to the 
nodal circles 

J.(ifcr) = (10) 

It is to be noticed that, owing to the equality of the frequencies of the two 
modes represented by (4), the normal modes are now to a certain extent 
indeterminate; viz. in place of cos«0 or Ansd we might substitute 
cos 8(0 — a,), where a, is arbitrary. The nodal diameters are then given by 

a 2m+l . . 

^ - a, = — 27~ ^' *• (^^) 

where m = 0, 1, 2, . . . , 5 — 1. The indeterminateness disappears, and the 
frequencies become unequal, if the boundary deviate, however slightly, from 
the circular form. 

In the case of the circular boundary, we obtain by superposition of two 
fundamental modes of the same period, in different phases, a solution 

S = C,J, (Jfcr) . cos (at T «fl + 6) (12) 

This represents a system of waves travelling unchanged round the origin 
with an angular velocity a/« in the positive or negative direction of 0. The 
motion of the individual particles is easily seen from Art. 189 (4) to be 
elliptic-harmonic, one principal axis of each elliptic orbit being along the 
radius vector. All this is in accordance with the general theory referred to 
in Art. 168. 

The most interesting modes of the unsymmetrical class are those corre- 
sponding to « = 1, e,g, 

t, = AJ-^ (hr) cos . cos (ai -h e), (13) 

where k is determined by 

J/(Jfca) = (14) 

The roots of this are* 

Jfca/7r='586, 1-697, 2-717*, (15) 

We have now one nodal diameter (6 = Jtt), whose position is, however, 
indeterminate, since the origin of is arbitrary. In the corresponding modes 
for an elliptic boundary, the nodal diameter would be fixed, viz. it would 
coincide with either the major or the minor axis, and the frequencies would 
be unequal. 

* See Bayleigh's treatise, Art. 339. A general formula for calculating the roots of J/ (£a) =0,. 
due to Prof. J. M«Mahon, is given by Gray and Mathews, p. 241. Numerioal tables are included 
in the collections of Dale, and Jahnke and Emde. 



280 



Tidal Waves 



[OHAP. vm 





191] Properties of BesseCs Functions 281 

The diagrams on the opposite page shew the contour-lines of the free 
surface in the first two modes of the present species. These lines meet 
the boundary at right angles, in conformity with the general boundary 
condition (Art. 190 (2)). The simple-harmonic vibrations of the individual 
particles take place in straight lines perpendicular to the contour-lines, 
by Art. 189 (4). The form of the sections of the free surface by planes 
through the axis of z is given by the curve y ^ J-i (x) on p. 278. 

The first of the two modes here figured has the longest period of all the 
normal types. In it, the water sways from side to side, much as in the 
slowest mode of a canal closed at both ends (Art. 178). In the second mode 
there is a nodal circle, whose radius is given by the lowest root of /^ (kr) = ; 
this makes r = •719a*. 

A oomparison of the preoeding investigation with the general theory of small oscilla- 
tions referred to in Art. 168 leads to several important properties of BesseVs Functions. 

In the first place, since the total mass of water is unaltered, we must have 

Ciir fa 

j I Crd0dr=O, (16) 

where ( has any one of the forms given by (4). For «>0 this is satisfied in virtue of the 
trigonometrical factor cos s3 or sin sB; in the symmetrical case it gives 



/: 



''jo{kr)rdr=0 (17) 





Again, since the most general free motion of the system can be obtained by superposi- 
tion of the normal modes, each with an arbitrary amplitude and epoch, it follows that any 
value whatever of {, which is subject to the condition (16), can be expanded in a series of 

the form 

f =22 {Ag cos 80 +B, sin sB) J, (kr) (18) 

where the summations embrace all integral values of a (including 0) and, for each value of 
s, all the roots k of (5). If the coefficients Ag, B« be regarded as functions of t, the equa- 
tion (18) may be regarded as giving the value of the surface-elevation at any instant. The 
quantities Ag, B, are then the normal co-ordinates of the present system (Art. 168) ; and in 
terms of them the formulae for the kinetic and potential energies must reduce to sums of 
squares. Taking, for example, the potential energy 

V =igp jjCdxdy, (19) 



f2w fa 
this requires that I f w^w^rdBdr^O^ 

J J 



(20) 



* The oacillatione of a liquid in a circular basin of any uniform depth were disouased by 
Poisson, "Sur lea petites oscillations de Teau contenue dans un cylindre/' Ann, de OergonnCf 
t. xix. p. 225 (1828-9); the theory of Bessers FunctionB had not at that date been worked out, 
and the results were consequently not interpreted. The full solution of the problem, with 
numerical details, was given independently by Rayleigh, Phil Mag. (5), t. L p. 257 (1876) [Papers, 
t. L p. 25]. 

The investigation in the text is limited, of course, to the case of a depth small in oomparison 
with the radius a. Poisson's and Bayleigh's solution for the case of finite depth will be noticed 
in Chapter ix. 



282 Tidal Waves [chap, vm 

where w^ , w^ are any two tenns of the expansion ( 18). UtOi.Wi involve coeines or sines of 
different multiples of 6, this is verified at once by integration with respect to B\ but if 
we take 

w^ oc t/« (hi r) cos sOf w^ x J. (^^'^) ^^^ '^» 

where hi, k^ are any two distinct roots of (5), we get 



/: 



a 



J. {k^r) J, (4,r) rrfr =0. (21) 

The general results, of which (17) and (21) are particular cases, are 

{''j^(kr)rdr=-^J^'(ka) (22) 

(cf. Art. 102 (10)), and 

/•« 1 

In the case of ki =^, the latter expression becomes indeterminate; the evaluation in the 
usual manner gives 

I* {J. (te)}« rdr =-~ [ik«a« {J/ (ka)}^ + (fc«a« - «») {J. (ka)}*] (24) 

For the analytical proofs of these formulae we refer to the treatises cited on p. 129. 

The small oscillations of an annular sheet of water bounded by concentric 
circles are easily treated, theoretically, with the help of Bessel's Functions * of 
the second kind.' The only case of any special interest, however, is when the 
two radii are nearly equal ; we then have practically a re-entrant canal, and 
the solution follows more simply by the method of Art. 178. 

The analysis can also be applied to the case of a circular sector of any 
angle*, or to a sheet of water bounded by two concentric circular arcs and 
two radii. 

192. As an example oi forced vibrations, let us suppose that the dis- 
turbing forces are such that the equilibrium elevation would be 



f=«©' 



cos 80 . COS (at + e) (25) 



This makes V^af = 0, so that the equation (12) of Art. 189 reduces to the form 
(1), above, and the solution is 

C = AJ, (*r) cos »e . cos (of + €), (26) 

where A is an arbitrary constant. The boundary-condition (Art. 189 (14)) 
gives 

AkaJ/ (ka) = sC, 

whence £ = C ^^^ ^ cos sB .cob (at + €) (27) 

The case « = 1 is interesting as corresponding to a uniform horizontal 
force; and the result may" be compared with that of Art. 179. 

♦ See Rayleigh, Theory of Sound, Art. 339. 



191-193] Basin of Variable Depth 283 

From the case 8 = 2 we could obtain a rough representation of the semi- 
diurnal tide in a polar basin bounded by a small circle of latitude, except that 
the rotation of the earth is not as yet taken into account. 

We notice that the expression for the amplitude of oscillation becomes 
infinite when J/ (ka) = 0. This is in accordance with a general principle, of 
which we have already had several examples ; the period of the disturbing 
force being now equal to that of one of the free modes investigated in the 
preceding Art. 

193*. When the sheet of water is of variable depth, the calculation at 
the beginning of Art. 189 gives, as the equation of continuity, 

dt" dx dy ^' 

The dynamical equations (Art. 189 (2)) are of course unaltered. Hence, 
eliminating ^, we find, for the free oscillations, 

ar^ = Haira^)^a^ravl ^'^ 

If the time-factor be «*<'<+•>, we obtain 

When h is a function of r, the distance from the origin, only, this may be 
written 

dhdC , <T« 
dr dr g 



l^^.K + T^ + ^-i-^ (4) 



As a simple example we may take the case of a oiroular basin which shelves gradually 
from the centre to the edge, according to the law 



=*.(!-.)• 



(6) 



Inteoduoing polar oo-ordinates, and assaming that ( yariee as cos »6 or sin a6, the equation 
(4) takee the form 

(^-a«,Ka;S+;:3^-^f)-J,'^+^f=0. (6) 

That integral of this equation which is finite at the origin is easily found in the form 
of an ascending series. Thus, assuming 

f'^^"©" (') 

where the trigonometrical factors are omitted, for shortness, the relation between consecu- 
tive coefficients is found to be 



(m«-««)^^ = |m(m-2)-^-^1^,,.„ 



* This formed Art. 189 of the 2nd ed. of this work (1896). A similar investigation is given 
by Poincar6, LeQona de mdcanique eOtaU, t. iil. (Paris, 1910). 



284 Tidal Waves [chap, vm 



o-»a« 



or, if we write -j— =n (n - 2) - «■, (8) 

where n is not as yet assumed to be integral, 

(m«-*«)24,„=(w-n)(m+»-2)24^., (9) 

The equation is therefore satisfied by a series of the form (7), beginning with the term 
At (r/ay, the succeeding coefficients being determined by putting m=«+2, «+4, ... in (0). 
We thus find 

A^A /''"V/i (n-s-2){n+8) f^ . (n-a-4)(n-^-2)(n+g)(n-fj+2) r* \ .^ 

^ 'W t 2(2«+2) ^^ 2. 4 (2« +2) (2^+4) a^-"'j'^'''> 

or in the usual notation of hypergeometric series 

f=^J..j(«./3,y.5) (11) 

where 0=^71+^8, /3 = 1+J«— Jn, y=« + l. 

Since these make y -a -/3=0, the series is not convergent for r=a, unless it terminate. 
This can only happen when n is integral, of the form a +2j. The corresponding values of 
<r are then given by (8). 

In the symmetrical modes (« =0) we have 

where j may be any integer greater than unity. It may be shewn that this expression 
vanishes for J —1 values of r between and a, indicating the existence oij—l nodal circles. 
The value of <r is given by 

<r«=4j(i-l)''^ (13) 

Thus the gravest symmetrical mode {j=2) has a nodal circle of radius -TOTa; and its 
frequency is determined by <r" = 8^Ao/^*' 

Of the unsymmetrical modes, the slowest, for any given value of 8, is that for which 

n =s« +2, in which case we have 

r* 
f =-4, — cos *tf cos (erf + €), 



the value of <r being given by o^=28 , ^ (14) 

The slowest mode of all is that for which 8 = 1, n =3; the free surface is then alwa3r8 
plane. It is found on comparison with Art. 191 (15) that the frequency is *768 of that of 
the corresponding mode in a circular basin of uniform depth h^, and of the same radius. 

As in Art. 192 we could at once write down the formula for the tidal motion produced 

by a uniform horizontal periodic force ; or, more generally, for the case where the disturbing 

potential is of the type 

Q ccr* cos 8$ cos {ai + c). 

194. We may conclude this discussion of Uong' waves on plane sheets 
of water by an examination of the mode of propagation of disturbances from 
a centre in an unlimited sheet of imiform depth. For simplicity, we will 
consider only the case of symmetry, where the elevation ^ is a function of 



193-194] BesseVs Function of the Second Kind 285 

the distance r from the origin of disturbance. This will introduce us to some 
peculiar and rather important features which attend wave-propagation in two 
dimensions. 

The investigation of a periodic disturbance involves the use of a Bessel's 
Function (of zero order) ' of the second kind,' as to which some preliminary 
notes may be useful. 

To solve the equation ^ '^~ d "^^'^ ^^^ 



by definite integrals, we assume* == f ^~'' ^^» 



(2) 



where 2^ is a function of the complex variable t, and the limits of integration are constants 
as yet unspecified. This makes 



^dz* 



by a partial integration. The equation (1) is accordingly satisfied by 

*=/ir;««l ('> 

provided the expression V(l+(")e-* 

vanish at each Umit of integration. Hence, on the supposition that z is real and positive, 
or at all events has its real part positive, the integral in (3) may be taken along a path 
joining any two of the points i, - 1, + oo in the plane of the variable t ; but two distinct 
paths joining the same points will not necessarily give the same result if they include 
between them one of the branch-points (t = ±:i) of the function under the integral sign. 

Thus, for example, we have the solution 



*^ = A 



>/(i+tr 



where the path is the portion of the imaginary axis which lies between the limits, and that 
value of the radical is taken which becomes = 1 for ^ =0. If we write ^ =£ +^17, we obtain 

<^ = »r J^J^j=2tJJ%08(2COS^)(W=»,rJo(2) (4) 

which is the solution already known (Art. 100). 

An independent solution is obtained if we take the integral (3) along the axis of 17 from 
the point (0, i) to the origin, and thence along the axis of £ to the point (00 > 0). This 
gives, with the same determination of the radical, 

fO e-*'^d(irf) r e-'U( _^ r e-'^dj _. f^e-''^dfj 
*""j< v/(l-'7») '^Josf{l+P)~)os^(l+$') *ioV('l-»7») 

By adopting other pairs of limits, and other paths, we can obtain other forms of <p, but 
these must all be equivalent to <f>i or <^2* ^^ ^ linear combinations of these. In particular, 
some other forms of 02 ^^ important. It is known that the value of the integral (3) taken 

* Forsyth, Differeniial Equations, c. vii. The systematic application of this method to the 
theory of BesseFs Functions is due to Hankel, "Die Cylinderfunktionen erster u. zweiter Art," 
Maik, Ann. t. i p. 467 (1869). See Gray and Mathews, 0. vii. 



(6) 



286 Tidal Waves [chap, vm 

round any closed contour which excludes the branch-points (t = ±^i) is zero. Let us first 

take as our contour a rectangle, two of whose sides 

coincide with the positive portions of the axes of £ 

and 7, except for a small semicircular indentation 

about the point t =t, whilst the remaining sides are 

at infinity. It is easily seen that the parts of the 

integral due to the infinitely distant sides wiU vanish, 

either through the vanishing of the factor c~*^ when f 

is infinite, or through the infinitely rapid fluctuation i^- 

of the fimction e'^'^/rj when 17 is infinite. Hence for 

the path which gave us (5) we may substitute that 

which extends along the axis of 17 from the point (0, i) 

to (0, too ), provided the continuity of the radical be 

attended to. Now as the variable t travels counter- ^ 

clockwise round the small semicircle, the radical 

changes continuously from V(l— »?*) *<> *V(»7*— !)• We have therefore 






(6) 



It will appear that this solution is the one which is specially appropriate to the case 
of diverging waves. Another method of obtaining it will be given in Chapter x. 

If we equate the imaginary parts of (5) and (6) we obtain 



a form due to Mehler*. 



2 r 

*^o (*) =- I si'i (^ cosh u) du, (7) 



On account of the physical importance of the solution (6) it is convenient to have a 
special notation for it. We writef 

Do(z)=- r«-<««»^»du (8) 

"■ y 

This is equivalent to Dq (z) =Zo (2) -tJ© (*)» W 

2 r 

wherej K^^ (z) =- I cos (z cosh u)du (10) 

^ y 

Equating the real parts of (5) and (6) we have, also, 

^0(2)=^- r «-"»"*» "rfi*-? f *' sin (z cos 5) (is (11) 

IF J »r y 

♦ Maih, Ann. t. v. (1872). 

t 2>o (z) is equivalent to - iH^^ (z) in the notation proposed by Nielsen. 

X As regards K^, this is the notation employed by Heine (except as to the constant factor), 
and H. Weber. The reader should be warned, however, that the same symbol has been employed 
in at least two other distinct senses in connection with the theory of BessePs Functions. 

From a purely mathematical point of view the choice of a standard solution *of the second 
kind' is largely a matter of convention, since the differential equation (1) is etill satisfied if we 
add any constant multiple of /q (z). In terms of the notation introduced by C. Neumann, which 
has found some acceptance, 

K, i") =1{-Yt (?) + (log 2 - y) J, (»)}. 

IT 

where 7 = '6772. . . (Enler*8 constant). 

A table of the function iirKQ (z) has been constructed by B. A. Smith ; see PhU. Mag. (5), 
t.xlv. p. 122(1898). 



194] BesseHs Function of the Second Kind 287 

For a like reason, the path adopted for ^^ may be replaced by the line drawn from the 
point (0» i) parallel to the axis of f (viz. the dotted line in the figure). To secure the con- 
tinuity of ^(\ -k-t^), we note that as i describes the lower quadrant of the small semicircle, 
the value of the radical changes from J(\ - 17') to e^**^ V(2£)> approximately. Hence along 
the dotted line we have, putting /=»+£, 

where that value of the radical is to be chosen which is real and positive when f is infini- 
tesimal Thus 

If we expand the binomial, and integrate term by term, we find 

^.,.,.(l)V..-*j..|^(-).!^(i)'....) ,.3, 

where use has been made of the formulae 

If we separate the real and imaginary parts of (13) we have, on comparison with (9), 

Jo (2) = (;^)* {5 sin (« + l»r) - >8f cos (2 + ifr) }, (16) 

Ko(z)={^^ {BQo%(z+\'!T)^89m(z+\fr)}, (16) 



/ 



(14) 






(17) 



, _ , 1«.3« 12.3*.6«.7* 

^*'^"' ^ = ^""2T(8i?"*' 4!(8z)* •• 

1« 1^3'. 5^ 

^"17(8^" 3!(8z)» """' 

The series in (13) and (17) are of the kind known as * semi-convergent,* or 'asymptotic,' 
expansions ; i.e. although for sufficiently large values of z the successive terms may for a 
while diminish, they ultimately increase again indefinitely, but if we stop at a small term 
we get an approximately correct result*. This may be established by an examination of 
the remainder after m terms in the process of evaluation of (12). 

It follows from (16) that the large roots of the equation Jy (z) =0 approximate to those of 

sm (z + Jir) -0 (18) 

The series in (13) gives ample information as to the demeanour of the function Do (z) 
when z is large. When z is small, D© (z) is very great, as appears from (9) and (11). An 
approximate formula for this case can be obtained as follows. Referring to (11), we have 

/;,— ..../%-K)^./;t±!{..i.'(i)-....)* 



■ll'^{'*s,*h{£}*-]*'- "?> 



* Gf. Whittaker, Modem Analysis^ c. viii. ; Bromwich, Theory of Infinite Series^ London 
1908, 0. xL The semi-convergent expansion of J^ (z) is due to Poisson, Joum, de V£c6U Polyt, 
cah. 19, p. 349 (1823) ; a rigorous investigation of this and other analogous expansions was given 
by Stokes, l.c, ante p. 277. The * remainder' was examined by lipschitz, CreUe, t. Ivi. p. 189 
(1869). Of. Hankel, Ic^ ante p. 286. 



288 Tidal Waves [chap, vin 

The first term gives* 

d«^=-y-logJz + ..., (20) 






and the remaining ones are small in comparison. Hence, by (9) and (11), 

^o(2)=--(logi2 + y + itV + ...) (21) 

IT 

2 
It follows that lim zDq (2) = - - f (22) 

8=0 ^ 

The formula (21) is sufficient for our purposes, but the complete expression can now be 
obtained by comparison with the general solution of (1) in terms of ascending series, viz. % 

Jo(2)logr + 22"*«2n«"^*«2n«76«"**7 ^^^^ 

1 1 1 

2"*"3'^*""^m' 



where «m = l +s + « + ...+- • 



In order to identify this with (21), for small values of z, we must make 

2 2 
5= , ^=--(logi+y+i»ir) (24) 

TT IT 

Hence 



2 2 fz' z^ z* 1 

^o(g)=~-(iogig+y+^*«')«^o(»)~- |22^"^«2^1«^'^^» 2^4^6« ~'•T 



(26) 



195. We can now proceed to the wave-problem stated at the beginning 
of Art. 194. For definiteness we will imagine the disturbance to be caused 
by a variable pressure p^ applied to the surface. On this supposition the 
dynamical equations near the beginning of Art. 189 are replaced by 

dt ^dx pdx' dt pdy p dy ' ^' 

if=-»(i+i) ,2, 

as before. 

If we introduce the velocity-potential in (1), we have, on integration, 

I-K + & (3) 

We may suppose that Pq refers to the change of pressure, and that the arbi- 
trary function of t which has been incorporated in <f) is chosen so that d<f>ldt = 

* De Morgan, Differential and Integral Calculus, London, 1842, p. 653. 

t The Beesers Functions of the second kind were first thoroughly investigated and made 
ayailable for the solution of physical problems in an arithmetically intelligible form by Stokes, 
in a series of papers published in the Camb. Trans. With the help of the modem Theory 
of Functions, some of the processes have been simplified by Lipschitz and others, and (especially 
from the physical point of view) by Rayleigh. These later methods have been freely used in 
the text. 

X Gray and Mathews, p. 11. 



194-195] Waves Diverging from a Centre 289 

in the regions not afiected by the disturbance. Eliminating { by means of 
(2), we have 

|^.^v.v + i|" W 

When ^ has been determined, the value of ^ is given by (3). 

We will now assume that f^ is sensible only over a small* area about the 
origin. If we multiply both sides of (4) by 8x8^, and integrate over the area 
in question, the term on the left-hand may be neglected (relatively), and we 
find 



'Pi^-jphilh^^y^ (^) 



gph dt, 

where hs is an element of the boundary of the area, and Sn refers to the hori- 
zontal normal to hs, drawn outwards. Hence the origin may be regarded as 
a two-dimensional source, of strength 

/("-^-ijt ■ <«) 

where Pq is the integral disturbing pressure. 

Turning to polar co-ordinates, we have to satisfy 

where c* = gh, subject to the condition 

,"^(-^l)=-^(^> («) 

where/ (i) is the strength of the source, as above defined. 

In the case of a simple-harmonic source e*'* the equation (7) takes the 
form 

'^ + i| + *V-o. (9) 

where k = a/c, and a solution is 

.^ = iZ)o(AT)6-*, (10) 

where the constant factor has been determined by Art. 194 (22). Taking the 
real part we have 

<f> = i {Kq (kr) cos at + Jq (^) sin aQ, (11) 

« 

corresponding to / (0 = ^^ ^^' 

* That is, the dimennons of the area are small compared with the * length' of the waves 
generated, this term being understood in the general sense of Art 172. On the other hand, 
the dimensions must be supposed large in compcuison with h» 

L.H. 19 



2»0 Tidal Waves [chap, vm 

For large values of hr the result (10) takes the form 

The combination t — rfc indicates that we have, in fact, obtained the solution 
appropriate to the representation of diverging waves. 

It appears that the amplitude of the annular waves ultimatdy varies 
inversely as the square root of the distance from the origin. 

196. The solution we have obtained for the case of a simple-harmonic 
source c*'* may be written 

•00 iff(t-- ooflh u ] 



2nr<f> = j 



e ' du (13) 





This suggests generalization by Fourier's Theorem ; thus the formula 



2^(f, = j fU^^coahujdu (U) 

should represent the disturbance due to a source /(^ at the origin*. It is 
implied that the form oif(t) must be such that the integral is convergent; 
this condition will as a matter of course be fulfilled whenever the source has 
been in action only for a finite time. A more complete formula, embracing 
both converging and diverging waves, is 

2nr<f)=j f(t--coBhu\du+j F (t + -cosh in du (15) 

The solution (15) may be verified, subject to certain conditions, by substitution in the 
differential equation (7). Taking the first term alone, we find 

= I •Isinh' u.f" (t — cosh tt J — cosh u.f'U — cosh « U dw 

This obviously vanishes whenever /(<)=0 for negative values of t exceeding a certain 
limitf. 

* The sahstanoe of Arts. 196 — 198 is adapted from a paper "On Wave-Propagation in Two 
Dimensions,*' Proc, Lond. Math. 8oc, t. xxxv. p. 141 (1902). A result equivalent to (14) was 
obtained (in a different manner) by Levi-Civita, Nuopo dmerUo (4), t. vi. (1897). 

t The verification is very similar to that given by Levi-Civita. 



1 95-1 97] Waves Diverging from a Centre 291 

Again, -2irr^=^/ ooBh«./M< — -coshttMu 

= ^| (8inhit+c-«)/' (<-%oshtt^dtt 

under the same condition. The limiting value .of this when r-»-0 is /(<); &nd the state- 
ment made above as to the strength of the source in (14) is accordingly verified. 

A similar process will apply to the second term of (15) provided F (t) vanishes for 
positive values of t exceeding a certain limit-. 

197. We may apply (14) to trace the effect of a temporary source varying 
according to some simple prescribed law. 

If we suppose that everything is quiescent until the instant t = 0, so that 
f{t) vanishes for negative values of i, we see from (14) or from the equivalent 
form 



27r^.= | 



J-^^l^ (16) 



that <f> will be zero everywhere so long as i < r/c. If, moreover, the source 
acts only for a finite time r, so that/ (t) = for i > r, we have, for <> r + r/c, 



P_0fli9 ,„)• 



This expression does not as a rule vanish ; the wave accordingly is not sharply 
defined in the rear, as it is in front, but has, on the contrary, a sort of 'tail''|' 
whose form, when < — r/c is large compared with r, is determined by 



27rA 1 ifVW<W (18) 



(<«*-f«/c*)' 
The elevation { at any point is given by (3), viz. 



£-^|-- m 



It follows that 



/ 



00 



Cdt = 0, (20) 



-00 



* Analytically, it may be noticed that the equation (4), when Po=^> °^7 ^ written 

and that (17) consiBtB of an aggregate of solutions of the known type 

t The existence of the 'tail' in the case of oylindrical electric waves was noted by Heaviside, 
PhU. Mag, (6), t. xxvi (1888) [Electrical Papers, t. ii], 

19—2 



292 Tidal Waves [chap, vm 

provided the initial and final values of ff> vanish. It may be shewn that this 
will be the case whenever /(f) is finite and the integral 



/ 



oo 



/(O* (21) 



— 00 



is convergent. The meaning of these conditions appears from (6). It 
follows that even when P© is always positive, so that the flux of liquid 
in the neighbourhood of the origin is altogether outwards, the wave which 
passes any point does not consist sblely of an elevation (as it would in the 
corresponding one-dimensional problem) but, in the simplest case, of an 
elevation followed by a depression. 

To illustrate the progress of a solitary wave we may assume 

/W = ?T^' (22) 

which makes P^ increase from one constant value to another according to the law 

Po = il + 5 tan-i - ( 23 ) 

The disturbing pressure has now no definite epoch of beginniiig or ending, but the range 
of time within which it is sensible can be made as small as we please by diminishing r. 
For purposes of calculation it is convenient to assume 

/W = ,-^ (24) 

in place of (22), and to retain in the end only the imaginary part. We have then 

2,.^=r_;^-*i — =2r-;^ — TT-^r ^^^ 

I t — coshtt-tV I t IT— (< + — ir\z^ 

Jo c Jo c \ c J 

I 

where z =:tanh ^u. We now write 

^_!:-»V=a«e-«^ t+--tr=6«e-«^ (26) 

c c 

where we may suppose that a, h are positive, and that the angles a, /3 lie between and \ir» 
Since 

)■ (27) 

tan2a=— — , tan 2/3 = -: — » 

it appeahf that'ct ^ !>*according as f ^ 0, and that a > /3 always. With this notation, we find 

2-^C 

To interpret the logarithms, let us mark, in the plane of a complex variable z, the points. 



197] 



SolUary Wave 



293 



Since the integral in the seoond member of (28) is to be taken along the path 01^ the proper 
value of the third member is 






((logg + t.OP/)-(logg-i.OG/)}, 




a 

where real logarithms and positive values of the angles are to be understood. Hence, 
rejecting all but the imaginary part, we find 

a . sin(a+/3), IP co8(a+/3), «,^, ,^, 

2^»= ab ^ ^gjQ^ ^;r~^(^-^^0) (29) 

as the solution corresponding to a source of the type (22). Here 

IP _ /a*+2ab cos (a +/3) +6«\i 2a6sin(a~ffl 
/0"V-2a6c6s(7->)+6V ' **^^^^- grZ^i (^) 

and the values of a, d, a» 3 in terms of r and t are to be found from (27). 

It will be sufficient to trace the effect of the most important part of the wave as it 
passes a point whose distance r from the origin is large compared with or. If we confine 
ourselves to times at which i —r/c is small compared with r/e, a will be small compared with 
6y PIQ will be a small angle, and IP/IQ will = 1, nearly. If we put 

< = -+Ttani;, (31) 

we shall have 

a=lir-i^» a = V(rseci7), /3=W» &=(2r/c)*, (32) 

approximately; and the formula (29) will reduce to 

2jr<^=^cosa=-~f^j cos (Jir - Ji;) V(C08 >?) (33) 




294 Tidal Waves [ohap- vm 

The elevation C la then given by 

approximately. The diagram on the preceding page shews the relation between ( and i^ as 
given by this formula*. 

198. We proceed to consider the case of a spherical sheet, or ocean, of 
water, covering a solid globe. We will suppose for the present that the globe 
does not rotate, and we will also in the first instance neglect the mutual 
attraction of the particles of the water. The mathematical conditions of the 
question are then exactly the same as in the acoustical problem of the 
vibrations of spherical layers of airf . 

Let a be the radius of the globe, h the depth of the fluid; we assume 
that h is small compared with a, but not (as yet) that it is imiform. The 
position of any point on the sheet being specified by the angular co-ordinates 
Oy ff>y. Jet. K .b^ the component velocity of the fluid at this point along the 
meridian, in the direction of 9 increasing, and v the component along the 
parallel of latitude, in the direction of ff> increasing. Also let ^ denote the 
elevation of the free surface above the nndisturbed level. The horizontal 

* ■ « 

motion being assumed, for the reasons explained in Art. 172, to be the same 
at all points in' a vertical line, the condition of continuity is 

^(wAasine&^)8& + ^(vAa8fl)8^ = ~a8ine&^.a8e.|^, 

where the left-hand side measures the flux out of the columnar space 
standing on the element of area a sin dh<f} . aSd, whilst the right-hand member 
expresses the rate of diminution of the volume of the contained fluid, owing 
to fall of the surface. Hence 



dt _ 1 ja (hu sin e) d (hv)\ 



f 



(1) 



dt a sin fl i dd ^ d<f> 

If we neglect terms of the second order in t^, t;, the dynamical equations 
are, on the same principles as in Arts. 169, 189, 

dt ^ add add' dt ^asinflg^ asin^a^' • ' ' '^ ' 

where Ci denotes the potential of the extraneous forces. 

If we put f = - Q/flf, (3) 

these may be written 

?^-._._«9^ „ 3^ asin^a^;^ ^^' ......W 

* The pointe marked - 1, 0« + 1 on the diagram correspond to the times r/c-T, rfCf rjc +r, 
respectively. 

t IMsoiissed in Rayleigh's Theory of Sound, c. zviii 



197-199] Wave% on a Spherical Sheet 29& 

Between (1) and (4) we can eluninate u, v, and bo obtain an equation in £ 
Only. 

In the case of simple-hannonic motion, the time-factor bdng e*<'*+•^ the 
equations take the forms 

y i j 9 (fusing) 8 {hv) ) .^. 

^""oasinflt dd ^ d<l> r '"^ ^ 

199. We will now consider more particularly the case of uniform depth. 
To find the free oscillations we put ^ = ; the equations (5) and (6) of the 
preceding Art. then lead to 

This is identical in form with the general equation of spherical surface^ 
harmonics (Art. 83 (2)). Hence, if we put 



a«a« 



, = w (n + 1), (2) 

a solution of (1) will be C = ^n> (3) 

where 8^ is the general surface-harmonic of order n. 

It was pointed out in Art. 86 that 8^ will not be finite over the whole 
sphere unless n be integral. Hence, for an ocean covering the whole globe, 
the form of the free surface at any instant is, in any fimdamental mode, that 
of a 'harmonic spheroid' 

r = a + A + iSn cos (<7« + €), (4) 

and the speed of the oscillation is given by 

a = {n(n+l)}*.^, (5) 

the value of n being integral. 



The characters of the various normal modes are best gathered from a 
study of the nodal lines (S„ = 0) of the free surface. Thus, it is shewn in 
treatises on Spherical Harmonics* that the zonal harmonic P„ (/n) vanishes 
for n real and distinct values of /x lying between ± 1, so that in this case 
we have n nodal circles of latitude. When n is odd one of these coincides 
with the equator. In the case of the tesseral harmonic 



a-'")"'^'rW. 



* For refeienceB see p. 103. 



20ft Tidal Waves [chap, vin 

the second factor vanishes for n — 8 values of /i, and the trigonometrical 
factor for 2a equidistant values of <f>. The nodal lines therefore consist of 
n — 8 parallels of latitude and 28 meridians. Similarly the sectorial harmonic 

has as nodal lines 2n meridians. 

These are, however, merely special cases, for since there are 2n 4- 1 
independent surface-harmonics of any integral order n, and since the 
frequency, determined by (5), is the same for each of these, there is a 
corresponding degree of indeterminateness in the normal modes, and in the 
configuration of the nodal lines. 

We can also, by superposition, build up various types of progressive 
waves; e.g. taking a sectorial harmonic we get a solution in which 

f « (1 - /x*)*** cos (tk^ - <rf + €) ; (6) 

this gives a series of meridianal ridges and furrows travelling round the 
globe, the velocity of propagation, as measured at the equator, being 

?-("-^)'-w* <') 

It 18 easily verified, on examination, that the orbits of the particles are now 
ellipses having their principal axes in the directions of the meridians and 
parallels, respectively. At the equator these ellipses reduce to straight 
lines. 

In the case n = 1, the harmonic is always of the zonal type. The 
harmonic spheroid (4) is then, to our order of approximation, a sphere 
excentric to the globe. It is important to remark, however, that this case 
is, strictly speaking, not included in our djmamical investigation, unless we 
imagine a constraint applied to the globe to keep it at rest; for the de- 
formation in question of the free surface would involve a displacement of 
the centre of mass of the ocean, and a consequent reaction on the globe. 
A corrected theory for the case where the globe is free could easily be 
investigated, but the matter is hardly important, first because in such a 
case as that of the Earth the inertia of the solid globe is so enormous 
compared with that of the ocean, and secondly because disturbing forces 
which can give rise to a deformation of the type in question do not as a 
rule present themselves in nature. It appears, for example, that the first 
term in the expression for the tide-generating potential of the sun or moon 
is a spherical harmonic of the 8ec<md order (see the Appendix to this 
Chapter). 

When n = 2, the free surface at any instant is approximately ellipsoidal. 
The corresponding period, as found from (5), is then -816 of that belonging 
to the analogous mode in an equatorial canal (Art. 181). 



199-^200] Free and Forced OmUations 297 

For large values of n the distance from one nodal line to another is 
small compared with the radius of the globe, and the oscillations then take 
place muchas OH a plane sheet of water. For example, the velocity of 
propagation, at the equator, of the sectorial waves represented by (6) tends 
with increasing n to the value (jrA)*, in agreement with Art. 170. 

From a compariiBOn of the foregoing investigation with the general theory of Art. 168 
we are led to infer, on physical grounds alone, the possibility of the expansion of any 
arbitrary value of f in a series of surface harmonics, thus 



00 





the coefficients of the various independent harmonics being the normal co-ordinates of the 
system. Again, since the products of these coefficients must disappear from the expressions 
for the kinetic and potential energies, we are led to the * conjugate' properties of spherical 
harmonics quoted in Art. 87. The actual calculation of the energies will be given in the 
next Chapter, in connection with an independent treatment of the same problem. 

The ejBEect of a simple-harmonic disturbing force can be written down at 

once from the formula (14) of Art. 168. If the surface value of Q be 

expanded in the form 

Q = SQ„, (8) 

where Q^ is a surface-harmonic of integral order n, the various terms are 
normal components of force, in the generalized sense of Art. 135; and the 
equilibrium value of f corresponding to any one term ii„ is 

Hence, for the forced oscillation due to this term, we have 

^• — T^'T' -^ 

where a measures the * speed' of the disturbing force, and a^ that of the 
corresponding free oscillation, as given by (5). There is no difficulty, of 
course, in deducing (10) directly from the equations of the preceding Art. 

200. We have up to this point neglected the mutual attraction of the 
parts of the liquid. In the case of an ocean covering the globe, and with 
such relations of density as we meet with in the actual earth and ocean, this 
is not insensible. To investigate its effect in the case of the free oscillations, 
we have only to substitute for Q„, in the last formula, the gravitation- 
potential of the displaced water. If the density of this be denoted by p, 
whilst po represents the mean density of the globe and liquid combined, we 
have* 

".=-£nf <") 

and 9 = iY^(^Po (12) 

* See, for example, Bouth, Analytical Statics, 2nd ed., Cambridge, 1902, t. ii pp. 146-7. 



298 Tidal Waves [qHAP, vm 

y denoting the gravitation-constant, whence 

"-=-2;nn[-^^^- -(i^i 

Substituting in (10) we find 

^^('-i^^ ("' 

where a^ is now used to denote the actual speed of the oscillation, and u^ 
the speed calculated on the former hypothesis of no mutual attraction. 
Hence the corrected speed is given by 



'■■-<"+')('-2;rTi£)S ('»' 



2n + 1 pj a^' 

For an ellipsoidal oscillation (n = 2), and for p/po = '18 (as in the case of 
the Earth), we find from (14) that the effect of the mutual attraction is to 
lotoer the frequency in the ratio of '94 to 1. 

Hie slowest oscillation would correspond to n = 1, but, as already indicated, 
it would be necessary, in this mode, to imagine a constraint applied to the 
globe to keep it at rest. This being premised, it appears from (15) that if 
p> Po the value of ci* is negative. The circular function of f is then replaced 
by real exponentials ; this shews that the configuration in which the surface 
of the sea is a sphere concentric with the globe is one of unstable equilibrium. 
Since the introduction of a constraint tends in the direction of stability, we 
infer that when p > po the equilibrium is a fortiori unstable when the globe 
is free. In the extreme case where the globe itself is supposed to have no 
gravitative power at all, it is obvious that the water, if disturbed, would tend 
ultimately, under the influence of dissipative forces, to collect itself into 
a spherical mass, the nucleus being expelled. 

It is obvious from Art. 168, or it may easily be verified independently, 
that the forced vibrations due to a given periodic disturbing force, when the 
gravitation of the water is taken into account, will be given by the formula 
(10), provided ii„ now denote the potential of the extraneous forces only, and 
cr„ have the value given by (15). 

201. The oscillations of a sea bounded by meridians, or parallels of 
latitude, or both, can also be treated by the same method f. The spherical 
harmonics involved are however, as a rule, no longer of integral order, and it 
is accordingly difficult to deduce numerical results. 

* This reanlt was given by Laplace, Micaniqut CAeste, livre l*", Art. 1 (1799). The tree and 
the forced oscillations of the tjrpe n=2 had been previously investigated in his "Becherohes snr 
quelques points du syst^me du monde/* M4m. de VAcad, roy, des ScienceSf 1775 [1778] [Oeuvre* 
Computes, t. ix. pp. 109, . . .]. 

t Of. Rayleigh, U. anU p. 294. . 



200^201] Waves on a Limited Ocean 299 

In the case of a zonal sea bounded by two parallels of latitude, we assnme 

C={ApW+Bq{^)}^js<l, (1) 

where fi^ooeS, and p (/i), q (fi) are the two fonctions of /t, containing (1 — /i*)^ as a factor, 
which are given by the formula (2) of Art. 86. It will be noticed that |> (/i) is an even, and 
q (fi) an odd function of /a. 

If we distinguish the limiting parallels by suffixes, the boundary conditions are that 
u =0 ioT fj, =s fi^ and fJL=fif For the free oscillations this gives, by Art. 198 (6), 



4p'(Mi)+5^(/*i)=0. ^p'(/*a)+5/(M«)=0 (2) 



whence 



=0. (3) 



P' (Ht)f ^' (h) 

which is the equation to determine the admissible values of n, the order of the harmonics. 
The speeds (o-) corresponding to the various roots are given as before by Art. 199 (5). 

If the two boundaries are equidistant from the equator, we have /x^ = —fi^ • The above 
solutions then break up into two groups ; viz. for one of these we have 

5=0, i>'(/ii)=0, (4) 

and for the other -4=0, 3^(fii)=0 (6) 

In the former case C has the same value at two points symmetrically situated on opposite 
sides of the equator; in the latter the values at these points are numerically equal, but 
opposite in sign. 

If we imagiue one of the boundaries to be contracted to a point (say /i^ = 1), we pass to 
the case of a circular basin. The values of ^' (1) and 9' (1) are infinite, but their ratio can 
be evaluated by means of formulae given in Art. 84. This gives, by the second of equations 
(2), the ratio A : B, and substituting in the first we get the equation to determine n. 
A simpler method of treating this case consists, however, in starting with a solution 
which is known to be finite, whatever the value of n, at the pole /i = l. This involves 
a change of variable, as to which there is some latitude of choice. 

We might take, for instance, the expression for P^* (cos 6) in Art. 86 (6), and seek to 
determine n from the condition that 



for^ = ^i*. 



^P,-(coe<?)=0 (6) 



By making the radius of the sphere infinite, we can pass to the plane problem of 
Art. 191 1« The steps of the transition will be understood from Art. 100. 

If the sheet of water considered have as boundaries two meridians (with or without 
parallels of latitude), say ^=0 and <^ =a, the condition that v=0 at these restricts us to 
the factor cos sa, and gives 8a =mir, where m is integral This determines the admissible 
values of «, which are not in general integral :(. 

* This question has been discussed by Maodonald, Proc Lond. Math. 8oe. t. xxzi. p. 264 
(1899). 

t Cf. Bayleigh, Theory of Sound, Arts. 336, 338. 

X The reader who wishes to carry the study of the problem farther in this direction is 
referred to Thomson and Tait, Nakiral PhUoiojJiy (2nd ed.). Appendix B, "Spherical Harmonic 
Analysis." 



300 Tidal Waves [chap* vm 

Tidal OsdUations of a Rotating Sheet of Water. 

202. The theory of the tides on an open sheet of water is seriously 
complicated by the fact of the earth's rotation. If, indeed, we could assume 
that the periods of the free oscillations, and of the disturbing forces, were 
small compared with a day, the preceding investigations would apply as 
a first approximation, but these conditions are far from being fulfilled in the 
actual circumstances of the earth. 

The difficulties which arise when we attempt to take the rotation into 
account have their origin in this, that a particle having a motion in latitude 
tends to keep its angular momentum about the earth's axis unchanged, and 
so to alter its motion in longitude. This point is of course familiar in 
connection with Hadley's theory of the trade- winds*. Its bearing on tidal 
theory seems to have been first recognised by Maclaurint. 

Owing to the enormous inertia of the solid body of the earth compared 
with that of the ocean, the ejSect of tidal reactions in producing periodic 
changes of the angular velocity is quite insensible. This angular velocity 
will therefore for the present be treated as constant :|:. 

The theory of the small oscillations of a dynamical system about a state 
of equilibrium relative to a solid body which rotates with constant angular 
velocity about a fixed axis difEers in some important particulars from the 
theory of small oscillations about a state of absolute equilibrium, of which 
some account was given in Art. 168. It is therefore worth while to devote 
a little space to it before entering on the consideration of special problems. 

203.' Let us take rectangular axes x, y, z fixed relatively to the solid, of 
which the axis of z coincides with the axis of rotation, and let <o be the angular 
velocity of the rotation. The equations of motion of a particle m relative to 
these moving axes are known to be 

w (^ - 2coy - w^x) = Z, m(y + 2ct}x- w^y) = 7, mz = Z, . .(1) 

where X, Y, Z are the impressed forces. From these we derive 

Swi {so Ax + yAy + i?Az) -f 2aiEm (xAy — yAx) 

- co»Sm (xAx -h yAy) = 2 {XAx + YAy + ZAz), . .(2) 

where the symbol A has the same meaning as in Art. 135. 

Let us now suppose that the relative co-ordinates (x, y, z) of each particle 

♦ "The Cause of the General Trade Winds," Phil. Trans, 1735. 

t l>e Causd Physied Fluxus et Rtftuxua Maris, Prop, yii : "Motus aqusa turbatnr ex inseqtiali 
velooitate qui corpora circa azem Terrne motu diomo defenintur** (1740). 

X The secviar effect of tidal friction in this respect will be noticed later (Chapter ix). 



202-203] Dynamics of a Rotating System 301 

can be expressed in terms of a certain number of independent quantities 
9i9 9i> ••• 9n9 ^^^ ^^^ '^ write 

® = iSm (A« + ya + i*), To = W^rn (x^ + y*) (3) 

Here ^ denotes the energy of the relative motion, which we shall suppose 
expressed as a homogeneous quadratic function of the generalized velocitiea 
9,., with coefficients which are functions of the generalized co-ordinates q^\ 
whilst Tq denotes the energy of the system when rotating with the solid, 
without relative motion, in the configuration (g^, 9t, ... 9n)* As ^ ^be 
proof of Lagrange's equations (cf. Art. 135) we find 

V /-A ^"A . :,A X 1^^ ^^\k^^(^^^ ^\ A 



2 



whilst 






a>«Sm(a^Ax + yAy) = ^Agi + ^Ag.+ ..,+|£"Ag (5> 



Also 2ci>Sw (^Ay - y Aa?) = (j3n?i + jSjj j, + • - • + Pmin) A?i 

+ (^tl?l + ^22?2 + . . • + ^2n?„) Agr, 
+ 

where j3,, = 2coSm //^' y\ , (7) 

and it is particularly to be noticed that 

Finally, we put 

S (ZAa? + YAy + ZAz) = ~ ^V -\' Q^liq^ + Q^^q^ + . . . + QnAg„, . .(9) 

where V is the potential energy, and Qj, Qj* ••• Qn are the generahzed 
comi)onents of disturbing force. 

If we substitute from (4), (5), (6) and (9) in (2), and equate separately 
the coefficients of Ajj, Ag2, • • • Ag„, we obtain n equations of the type* 

It may be noticed that these equations may be obtained as a particular 
case of Art. 141 (24), with the help of Art. 142 (8), by supposing the rotating 
solid to be free, but to have an infinite moment of inertia. 

< 

* Cf. Thomson and Tait, Natural PhUoaophy (2nd ed.), Part i. p. 319. 



802 Tidal Waves [chap, vra 

The conditions for relative equiUbrium, in the absence of disturbing 
forces, are found by putting ji, 92* • • • jn = ^ ii^ {10)> ^^ more simply from (2). 
In either way we obtain 

|-(F-!ro) = 0. (11) 

1 

which shews that the equilibrium value of the expression 7 — T© is 
'stationary.' 

If T denote the total kinetic energy of the system, we have 
T = \i:m{(x - a>y)« + (y + o^xf + i«} = ® + To + ^Sm (xy - yx), . .(12) 
whence, on reference to (1), 

J(r4-7) = |(3r+ro+ F) + coSm(a;y-y^) 

= |(®+ F-ro) + ^s(a?y-y^) (13) 

This is to be equated to the rate at which the disturbing forces do work, 
i.e. to 

a>S (xY - yX) + Qi9i + Qjft + . . . + Qnin^ 

Hence | (® + ^ "^ ^0) = OWi + ^292 + • • • + (?n?n (14) 

This result may also be deduced from the equations (10). 

When there are no disturbing forces, we have 

3r+ 7-^0 = const (15) 

It may be noticed that the analogue of the Hamiltonian equation (20) of 
Art. 135 is now 

f '[A{®+ ro + coSw(»y-yic)} + S(ZAx+ yAy + ZA2)]cft = 0, ..(16) 

with the condition that the variations Ax, Ay, Az of the rdaJtive co-ordinates 
must vanish at both limits. This follows easily from (1). If we transform 
to generalized co-ordinates, using (9), we may derive an independent proof 
of the equations (10). 

204. We will now suppose the co-ordinates q^ to be chosen so as to vanish 
in the undisturbed state. In the case of a amaJl disturbance, we may then 
write 

2® = Oii^i* + a^q^ + . . . + 2ai2?i?2 + (1) 

2(7- To) " Ciig* + ^22?' + . . . + 2cx2gi?2 + • • . , (2) 

where the coefficients may be treated as constants. The terms of the 
first degree in 7 — T© ^*ve been omitted, on account of the 'stationary' 
property. 



203-206] Principal Co-ordinates 303 

In order to simplify the equations as much as possible^ we will further 
suppose that, hj a linear transformation, each of these expressions is reduced, 
as in Art. 168, to a sum of squares ; viz. 

23r = ai?i" + a2?2" + . . - + Mn*, (3) 

, 2 (7 - To) - CiJi« + c^qj" + • . . + c^qn^ (4) 

The quantities 9i, jt, ... qn niay be called the 'principal co-ordinates' of the 
system, but we must be on our guard against assuming that the same 
simplicity of properties attaches to them as in the case of no rotation. The 
coefficients o^, a,, ... a„ and c^, Cg, ... c,, may be called the 'principal 
coefficients' of inertia and of stability, respectively. The latter coefficients 
are the same as if we were to ignore the rotation, and to introduce fictitious 
'centrifugal' forces (nuo^x, mwh/, 0) acting on each particle in the direction 
outwards from the axis. 

The equations ^10) of the preceding Art. become, in the case of infinitely 
small motions, 

«i§i + ^i?i + Pnqt + ftsft + . . . -f Pmqn = Qly 

«2?i + cjja + Ptiqi + Puqz + . . . + Ptnqn = Qi, [ (5) 

«n?n+ <^nqn-^ ^nl?l + ^fil?2 + ^n8?8 + • • • = Qnf 

where the coefficients Prs ^^7 ^^ regarded as constants. 

If w:e multiply these hj qi, qt, ... ?» ^ order and add, we find, taking 
account of the relation j5„ = — ftr* 

|(®+l'-2'o) = ei?l + e2?2+...+(?n?n, (6) 

as has already been proved without approximation. 

205. To investigate the /re6 motions of the system, we put Qi, Qt, ... 
Qn = Oyia (5), and assume, in accordance with the usual method of treating 
linear equations, 

q^^A^e^^ qt^A^^\ ...q^^A^^* (7) 

Substituting, we find 

(aiA« + Ci) A^ + p^Mt + . . . + AnA^n = 0, 

jSni A^i + j5h, Ail, + . . . + K A» + cj An = 0. 

Eliminating the ratios Aii A^i ... : Af^,we get the equation 

a^A* + Ci, Pi^X, . . . ^i«A 

ftiA, agA* + Cj, ... ^t«A 



^nlA, ^n2A, ... anA* + C, 



= 0, (9) 



304 Tidal Waves [chap, vin 

or, as we shall occasionally write it, for shortness, . 

A(A) = (10) 

The determinant A (A) comes under the class called by Cayley 'skew- 
determinants,* in virtue of the relations (8) of Art. 203. If we reverse the 
sign of A, the rows and columns are simply interchanged, and the value of the 
determinant is therefore unaltered. Hence, when expanded, the equation (9) 
will involve only even powers of A, and the roots will be in pairs of the form 

A = ± (p -+• i(T). 

In order that the configuration of relative equilibrium should be stable 
it is essential that the values of p should all be zero, for otherwise terms of 
the forms e*^* cos at and e**** sin at would present themselves in the realized 
expression for any co-ordinate q^. This would indicate the possibility of an 

oscillation of continually increasing amplitude. 

• 

In the theory of absolute equilibrium, sketched in Art. 168, the necessary 
and sufficient condition of stability (in the above sense) was simply that the 
potential energy must be a minimum in the configuration of equilibrium. In 
the present case the conditions are more complicated*, but it is easily 
seen that if the expression for V — T^ be essentially positive, in other words 
if the coefficients c^, Cj, . . . c^ in (4) be all positive, the equilibrium must be 
stable. This follows at once from the equation 

® + (F - To) - const., (11) 

proved in Art. 203, which shews that under the present supposition neither 
© nor V — Tq can increase beyond a certain limit depending on the initial 
circumstances. It will be observed that this argument does not involve 
the use of approximate equations f. 

Hence stability is assured if F — Tq is a minimum in the configuration 
of relative equilibrium. But this condition is not essential, and there may 
even be stability (from the present point of view) with F — T© * maximum, 
as will be shewn presently in the particular case of two degrees of freedom. 
It is to be remarked, however, that if the system be subject to dissipative forces, 
however slight, afEecting the relative co-ordinates g^, q^^ ... g„, the equi- 
librium will be permanently or * secularly ' stable only if F — Tq is a minimum. 
It is the characteristic of such forces that the work done by them on the 
system is always negative. Hence by (6) the expression 2tH- ( F. — Tq) will, so 
long as there is any relative motion of the system, continually diminish, in 
the algebraical sense. Hence if the system be started from relative rest in a 

* They have been investigated by Routh, On ike Stability of a Given State of Motion; Bee 
also his Advanced Rigid Dynamics, c. vi. 

t The argument was originally applied to the theory of oscillations about a configuration of 
absolute equilibrium (Art. 168) by Dirichlet, "Ueber die Stabilit&t des Gleichgewichts," CrelUf 
t. xxzii. (1846) [Werke, Berlin, 188^97, t ii p. 3]. 



205] Ordinary and Seeidar Stability 305 

configuration such that V — T^^'iB negative, the above expression, and therefore 
A fortiori the part F — To* ^^ assume continually increasing negative values, 
which can only take place by the system deviating more and more from its 
equilibrium-configuration. 

This important distinction between * ordinary' or kinetic, and ^seculat* 
or practical stability was first pointed out by Thomson and Tait*. It is to 
be observed that the above investigation presupposes a constant angular 
velocity (a>) maintained, if necessary, by a proper application of force to the 
rotating solid. When the solid is free, the condition of secular stability takes 
a somewhat different form, to be referred to later (Chapter xii.). 

To examine the character of a free oscillation, in the case of stability, we 
remark that if A be any root of (10), the equations (8) give 

A,r(A) A«(A)-A«(A) A„(A) "" ^'^^ 

where A^i, A^, A^j, . . . Ay„ are the minors of any row in the determinant A, 
and C is arbitrary. It is to be noticed that these minors will as a rule involve 
odd as well as even powers of A, and so assume unequal values for the two 
oppositely signed roots (± A) of any pair. If we put A = ± ia, the general 
symbolical value of g, corresponding to any such pair of roots may be 
written 

q, = C^rs (^) e*^ -f C'A„ (- to) e-^. 
If we put 2A„ (icr) = F, (cr^) + iof. [p% 

C = K^, C = fCe-**, 

we get a solution of our equations in real form, involving two arbitrary 
constants iC, c; thusf 

q^ = F^ {a^) . K cos (at + c) - cr/i (a^) . K sin (at + c), 

q^ = F^ (a^) . K cos (at + e) - af^ (a*) . K sin (at + c), 

?8 = ^3 (o^) . K cos (at + e)- af^ (a^) . K sin (at + c), ^ 



(13) 



g„ = jF„ (a^) . K cos (at + c) - af^ (a») . K sin (at + c). 

These formulae express what may be called a * natural mode' of oscillation 
of the system. The number of such possible modes is of course equal to the 
num]ber of pairs of roots of (10), i.e. to the number of degrees of freedom of 
the system. 

♦ Natural Philosophy (2nd ed.), Part i. p. 391. See also Poincar^, "Sur T^quilibre d*une 
masse (luide anim^e d'un mouvement de rotation," Acta Mathematical t. vii. (1885), and op. eit 
ante p. 141. 

t We might have obtained the same result by assuming, in (6), 

where A, is real, and rejecting, in the end, the imaginary parts. 

L. H. 20 



306 



Tidal Waves 



[chap, vm 



If ^, 1), ^ denote the component displacements of any particle from its 
equilibrium position, we have 



J. dx , dx 
dy dy 



, dx 
dz 



(14) 



y dz dz 



Substituting from (13), we obtain a result of the form 

^ = P . fC cos (a« + c) + P' . fC sin (at + c), ] 

1) =- g . Z cos (cr« + €) + e' . fC sin (a« + €), i (15) 

i = R.K COB (at + €) -\' R' .K sin (at + c), J 

where P, P', Q, Q\ 22, 22' are determinate functions of the mean position of 
the particle, involving also the value of a, and therefore dijSerent for the 
difEerent normal modes, but independent of the arbitrary constants K, e. 
These formulae represent an elliptic-harmonic motion of period in/a, the 
directions 






and t^IL-1 
P' Q'~R" 



• •••••••*• \a.\9 I 



being those of two conjugate semi-diameters of the elliptic orbit, of lengths 

(P2 ^ Q2 + jj2)i . X, and (P'a + g'^ ^ 22'«)* . If, 

respectively. The positions and forms and relative dimensions of the elliptic 
orbits, as well as the relative phases of the particles in them, are accordingly 
in each natural mode determinate, the absolute dimensions and epochs being 
alone arbitrary *. 

206. The symbolical expressions for the forced oscillations due to a 
periodic disturbing force can easily be written down. If we assume that 
Qi9 Q^y - " Qn ^^ vary as e*^, where a is prescribed, the equations (5) give, 
if we omit the time-factors. 



An(tcr)^ . Ai2(*or)^ 
^^ A (ta) ^ A (ta) 

A21 (ia ) ^ , A^Mn A. 



^(^a) 



^(^a) 



■^ A{ia) ^"' 



Anl (^y) ^ , K2 (^) n A. A^^nn (^) 



A (ia) 



A(ia) 



^(^a) 



Qn^ 



(17) 



* The theory of the free modes has been further developed by Rayleigh, "On the Free 
Vibrations of Systems affected with Small Rotatory Terms," Phil. Mag. (6), t. v. p. 293 (1903) 
[Papers, t. v. p. 89], for the case where the rotatory coefficients /S^, are relatively small. 



205-206] Free and Forced OsciUatiom 307 

The most important point of contrast with the theory of the 'normal 
modes' in the case of no rotation is that the displacement of any one type is 
no longer affected solely by the disturbing force of that type. As a con- 
sequence, the motions of the individual particles are, as is easily seen from 
(14), now in general elliptic-harmonic. Again, there are in general differences 
of phase, variable with the frequency, between the displacements and the force. 

As in Art. 168, the displacement becomes very great when A (ia) is very 
small, i.e. whenever the 'speed' a of the disturbing force approximates to 
that of one of the natural modes of free oscillation. 

When the period of the disturbing forces is infinitely long, the displace- 
ments tend to the * equilibrium- values ' 

?i — T"> ?2 ——>••• ?n — —•, lio) 

as is found by putting or = in (17), or more simply from the fundamental 
equations (5). This conclusion must be modified, however, when one or 
more of the coefficients of stability c^, Cj, . . . c„ is zero. If, for example, 
^1 = 0, the first row and column of the determinant A (A) are both divisible 
by A, so that the determinantal equation (10) has a pair of zero roots. In 
other words we have a possible free motion of infinitely long period. The 
coefficients of Q^y Q3, ... Qn ^^ ^^^ right-hand side of (17) then become 
indeterminate for a ~ 0, and the evaluated results do not as a rule coincide 
with (18). This point is of importance, because in some hydrodynamical 
applications, as we shall see, steady circulatory motions of the fluid, with 
a constant deformation of the free surface, are possible when no extraneous 
forces act; and as a consequence forced tidal oscillations of long period do 
not necessarily approximate to the values given by the equilibrium theory of 
the tides. Cf. Arts. 214, 217. 

In order to elucidate the foregoing statements we may consider more in detail the case 
of two degrees of freedom. The equations of motion are then of the forms 

<hSi +^1^1 +^J8 =Qi, (hit +c,g, -/9^i =^2 (10) 

The equation determining the periods of the free oscillations is 

OjajX* +(aiC3| +a^Ci +/3*) X^+CjC, =0. (20) 

For 'ordinary' stability it is sufficient that the roots of this quadratic in X' should be real 
and negative. Since £4, o^ are essentially positive, it is easily seen that this condition is 
in any case fulfilled if c^, c, ^^^ both positive, and that it will also be satisfied even when 
Cj , C2 are both negative, provided iS* be sufficiently great. It will be shewn later, however, 
that in the latter ca«e the equilibrium is rendered unstable by the introduction of 
dissipative forces. See Art. 316. 

To find the forced oscillations when Qi, Q2 vb^ as e**^, we have, omitting the 
time-factor, 

{Ci-a^ih) gi +i<r^2 =Qi» 'i<^^i +(©2 "O-'^j) gj=Qt» (21) 

whence a - ' i^-<^<h)Qi'i<rfiQ2 . *V^Oi + (q -cr»a,) Q. 

Whence ^1 -(c, _^t^) (c, -<,««,) -<r«i3»' ^»"(c, -<r«ai)(c,-<r»a,)-a*/3« ^^^^ 

20—2 



308 Tidal Waves [chap, viii 

Let us now suppose that c, =0, or, in other words, that the displacement q^ does not 
affect the value of V —T^, We will also suppose that Qt=0, i.e. that the extraneous 
forces do no work during a displacement of the type q^' The above formulae then give 

In the case of a disturbance of long period we have o- =0, approximately, and therefore 

'-^T^«- *«=5;^«» <^) 

The displacement q^ is therefore has than its equilibrium- value, in the ratio 1 : 1 +/3V^^ > 
and it is accompanied by a motion of the type q^ although there is no extraneous force of 
the latter type (cf. Art. 217). We pass, of course, to the case of absolute equilibrium, 
considered in Art, 168, by putting /9=0*. 

It should be added that the determination of the * principal co-ordinates* 
of Art. 204 depends on the original forms of ST and V — Tq, and is therefore 
afEected by the value of a>*, which enters as a factor of Tq* The system of 
equations there given is accordingly not altogether suitable for a discussion 
of the question how the character and the frequencies of the respective 
principal modes of free vibration vary with a>. One remarkable point which 
is thus overlooked is that types of circulatory motion, which are of infinitely 
long period in the case of no rotation, may be converted by the sUghtest 
degree of rotation into oscillatory modes of periods comparable with that of 
the rotation. Cf. Arts. 212, 223. 

To illustrate the matter in its simplest form, we may take the case of two degrees of 
freedom. If c, vanishes for a> =0, and so contains co' as a factor in the general case, the 
two roots of equation (20) are 

approximately, when o>' is smalL The latter root makes X x », ultimately. 

207. Proceeding to the hydrodynamical examples, we begin with the 
case of a plane horizontal sheet of water having in the undisturbed state a 
motion of uniform rotation about a vertical axisf. The results will apply 
without serious qualification to the case of a polar or other basin, of not too 
great dimensions, on a rotating globe. 

Let the axis of rotation be taken as axis of z. The axes of x and y being 
now supposed to rotate in their own plane with the prescribed angular 
velocity co, let us denote by w, v, w the velocities at time t, relative to these axeSy 
of the particle which then occupies the position (x, y, z). The actual velocities 
of the same particle, parallel to the instantaneous positions of the axes, will 
be w — oiy, V -f wx, w, and the accelerations in the same directions will be 

Di - ^^^ - ^ ^' 5^ + ^^ - ^ y' -Dt' 

* The preceding theory appeared in the 2nd ed. (1896) of this work. 

t Sir W. Thomson, *'0n Gravitational Oscillations of Rotating Water," Proc, R, 8. Edin^ 
t. X. p. 92 (1879) [Papere, t. iv. p. 141], 



206-207] Two-Dimemional Problems 309 

In the present application^ the relative motion is assumed to be infinitely 
small, so that we may replace D/Dt by d/dt. 

Now let Zq be the ordinate of the free surface when there is relative 
equilibrium under gravity alone, so that 

CO* 

2o = i— (^* + y*) H- const., (1) 

if 

as in Art. 26. For simplicity we will suppose that the slope of this surface 
is everywhere very small ; in other words, if r be the greatest distance of any 
part of the sheet from the axis of rotation, a)^r/g is assumed to be small. 

If Zq + ( denote the ordinate of the free surface when disturbed, then on 
the usual assumption that the vertical acceleration of the water is small 
compared with g, the pressure at any point (x, y, z) will be given by 

y-yo==^p(2^o+?-2), (2) 

The equations of horizontal motion are therefore 

__2a«,= -s,g^-g-, ^ + 2a,«=-<7^--^, ....(3) 
where Q denotes the potential of the disturbing forces. 

* 

If we write f « , (4) 

9 

these become 

3|-2a«,= -4(C-r). |+2a,«=-i,|(C-0. ..(5) 
The equation of continuity has the same form as in Art. 193, viz. 

dt" dx dy ' ^' 

where h denotes the depth, from the free surface to the bottom, in the 
undisturbed condition. This depth will not, of course, be uniform unless the 
bottom follows the curvature of the free surface as given by (1). 

If we eliminate C^Cfrom the equations (5), by cross-differentiation, we find 

di[^-dj,)*^[^^^)=^' <'' 

or, writing u^d(/dt, v=dri/dt, and integrating with respect to t, 

£-5*^(3*1)-"°^ "' 

This 18 merely the expression of Helmholts' theorem that the product of the vortioity 

2» + ^ - R- and the cross-section 1 1 +^ + 5^) ^^y» 
of a vortex -filament, ib constant. 



310 Tidal Waves [chap, vm 

In the case of a simple-harmonic disturbance, the time-factor being 6^*, 
the equations (5) and (6) become 

icm-2cot;=-(7^U~f), iat; + 2a>ti«-(7|^(C-?), ..(9) 

J . y 3 (Aw) 9 (At?) ,-^. 

*^^ ^^^--k---V (^^^ 

From (9) we find 

(11) 

and if we substitute from these in (10), we obtain an equation in ( only. 
In the case of uniform depth the result takes the form 

Vi*$ + '^^*^ = V,«f, (12) 

where Vj* = d^jdx^ + a«/9y*, as before. 

When ^=0, the equationfi (6) and (6) can be satisfied by coMkmi values of u, v, C 
provided certain oonditions are fulfilled. We must have 

'*"""2i^' *'-2iS ^^^^ 

andtherefore ^f^' ^ =0 (14) 

9 («, y) 

The latter condition shews that the contour-lines of the free surface must be everywhere 
parallel to the contour-lines of the bottom, but that the value of ( is otherwise arbitrary. 
The flow of the fluid is everywhere parallel to the contour-lines, and it is therefore 
further necessary for the possibility of such steady motions that the depth should be 
uniform along the boundary (supposed to be a vertical wall). When the depth is every- 
where the same, the condition (14) is satisfied identically, and the only limitation on the 
vcdue of ( is that it should be constant along the boundary. 

208. A simple application of the preceding equations is to the case of 
free waves in an infinitely long uniform straight canal*. 

Ifweassnme j «= ^6** '«*-*> +"»^ v = 0, (1) 

the axis of x being parallel to the length of the canal, the equations (9) of 
the preceding Art., with the terms in f omitted, give 

cu^gi, 2a}U^-gmC, (2) 

whilst, from the equation of continuity (Art. 207 (6)), 

c^^hu (3) 

We thence derive c^=: gh, m = — 2ct>/c (4) 

* Sir W. ThomBon, U, ante p. SOS 



207-209] Circular Basin 311 

The former of these results shews that the wave- velocity is unaffected by the 
rotation. 

When expressed in real form, the value of t, is 

I = ae-^yl^ cos {* (c« - a;) + c} (5) 

The exponential factor indicates that the wave-height increases as we 
pass from one side of the canal to the other, being least on the side which 
is foTwardy in respect of the rotation. If we take account of the directions 
of motion of a water-particle, at a crest and at a trough, respectively, this 
result is easily seen to be in accordance with the tendency pointed out in 
Art. 202*. 

The problem of determining the free oscillations in a rotating canal of 
finite length, or in a rotating rectangular sheet of water, has not yet been 
solved f. 

209. We take next the case of a circular sheet of water rotating about 
its centre t. 

If we introduce polar co-ordinates r, fl, and employ the symbols fi, to 
denote displacements along and perpendicular to the radius vector, then since 
R = iafi, = ia%y the equations (9) of Art. 207 are equivalent to 

whilst the equation of continuity (10) becomes 

d(hRr) d(h%) 

*"" rdr rde ^^' 

Hence 

^^^ - w (a; " Vfae) ^^ ~ ^^' ® ° a> -"w Wd^ ~ *i^) ^^ ~ ^' 

(3) 

and substituting in (2) we get the differential equation in £. 

In the case of uniform depth we find 

( V + K«) J = V,»^, (4) 

-<i ''*=^^' (') 

This might have been written down at once from Art. 207 (12). 

* For applications to tidal phenomena see Sir W. Thomson, Nature, t. xiz. pp. 154, 671 (1879). 

t Except in the case where the angular velocity of rotation is relatively small. For this see 
Rayleigh, "On the Vibrations of a Rectangular Sheet of Rotating Liquid," Phil Mag. (6), t. v. 
p. 297 (1903) [Papers, t. v. p. 93]. 

X The investigation which follows is a development of some indications given by Kelvin in 
the paper cited on p. 308. 



312 Tidal Waves [chap, vra 

The condition to be satisfied at the boundary (r = o, say) is 22 = 0, or 

a 2to a 



('!—»)«-£'-» <" 



210.- In the case of the free oscillations we have f = 0. The way in 
which the imaginary i enters into the above equations, taken in conjunction 
with Fourier's Theorem, suggests that occurs in the form of a factor 6***, 
where 8 is integral. On this supposition, the differential equation (4) becomes 



dr* 



^Yhi^-T^i-" <«) 



and the boundary-condition (7) gives 

4^^C = 0. ..(9) 

for r ^ a. 

The equation (8) is of Bessel's form, and the solution which is finite for 
r = may therefore be written 

i = AJ, (kt) e'^^*+^ ; (10) 

but it .is to be noticed that k^ is not, in the present problem, necessarily 
positive. When k* is negative, we may replace J. {kt) by Z, (/cV), where 
K is the positive square root of (4co* — <^)l9K and 

• ^^^ " 2* . » ! 1 ^ 2 (2« + 2) ^ 2 . 4 (2« + 2) (2s + 4) "^ • • • J * "^^ 

In the case of symmetry about the axis (s = 0), we have, in real form, 

^ = ^JoM-cos(a« + €), (12) 

where k is determined by 

Jo'('ca) = (13) 

The corresponding values of a are then given by (6). The free surface has, 
in the various modes, the same forms as in Art. 191, but the frequencies are 
now greater, viz. we have 

(7« = V + 4a>«, (14) 

where Oq is the corresponding value of a when there is no rotation. It is 
easily seen, moreover, on reference to (3), that the relative motions of the fluid 
particles are no longer purely radial ; the particles describe, in fact, ellipses 
whose major axes are in the direction of the radius vector. 

For « > we have 

i = AJ^ (kt) . cos (or« + «& + €), (15) 

where the admissible values of /c, and thence of a, are determined by (9), 
which gives 

koJ: (ko) + — J, {Ka) = (16) 

* The fonotions /, (2) were tabulated by Prof. A Lodge, Brit Asa, Rep. 1889. The tables are 
reprinted by Dale, and by Jahnke and Emde. 



209-210] 



Circular Basin 



313 



The fonniila (15) represents a wave rotating relativdy to the water with 
an angular velocity a/«, the rotation of the wave being in the same direction 
with that of the water, or the opposite, according as a/a> is negative or 
positive. 

Some indioatioDB as to the values of <r may be gathered from a graphical oonstruction. 
If we write jc'a' =a?, we have, from (6), 

^=4^1)*' ^"^ 



where 

If we farther put 

the equation (16) may be written 



3 = 



icaJ/ (ko) 



gh 



=<t>{ic^a*). 



.(18) 



<l>{x)± 



(-1)*- 



.(19) 
.(20) 



The curve y = - ^ (a?) 

can be readily traced by means of the tables of the functions Jg (z), /« (z) ; and its inter- 
sections with the parabola 

y« = l+ar//3 ,.(21) 

will give, by their ordinates, the values of cr/2oi>. The constant jS, on which the positions 
of the roots depend, is equal to the square of the ratio 2<oal{gh)^ which the period of 
a wave travelling roimd a circular canaJ of depth h and perimeter 2ira bears to the 
half -period {ir/») of the rotation of the water. 

The accompanying figures indicate the relative magnitudes of the lower roots, in the 
oases « = 1 and 8=2, when /9 has the values 2, 6, 40, respectively*. 

p.2 




* For clearness the scale of y has been taken to be 10 times that of z. 



314 



Tidal Waves 



[CHAP, vin 



With the help jof these figures we can trace, in a general way, the changes in the 
character of the fiee modes as /9 increases from zero. The results may be interpreted aa 
due either to a continuous increase of «, or to a continuous diminution of A. We will use 




[.=2] 



the terms 'positive' and 'negative' to distinguish waves which travel, relatively to the 
water, in the same direction as the rotation and the opposite. 

When /3 is infinitely small, the values of x are given by J, (x^) =0; these correspond 
to the vertical asymptotes of the curve (20). The values of o- then occur in pairs of 
equal and oppositely-signed quantities, indicating that there is now no difference between 
the velocity of positive and negative waves. The case is, in fact, that of Art. 191 (12). 

As jS increcbses, the two values of <r forming a pair become unequal in magnitude, and 
the corresponding values of x separate, that being the greater for which (r/2ci> is positive. 
When /3 =«(« + !) the curve (20) and the parabola (21) Umch at the point (0, — 1)» 
the corresponding value of o- being —2», As fi increases beyond this critical value^ 
one value of x becomes negative, and the corresponding (negative) value of cr/2» becomea 
smaller and smaller. 

Hence, as jS increases from zero, the relative angular velocity becomes greater for a 
negative than for a positive wave of (approximately) the same type; moreover the value 
of (T for a negative wave is always greater than 2«. As the rotation increases, the two 
kinds of wave become more and more distinct in character as well as in 'speed.' With a 
sufficiently great value of /3 we may have one, but never more than one, positive wave for 
which cr is numerically less than 2a>. Finally, when /3 is very great, the value of o- 
corresponding to this wave becomes very small compared with «, whilst the remaining 
values tend all to become more and more nearly equal to ±2». 



210-211] Circular Basin 315 

If we use a zero suffix to distinguish the ease of <o =0, we find 

<r«_ ic'+4a)VyA _a;+/3 

_ "";:"» y^^f 

VQ <0 *0 

where x^ refers to the proper asymptote of the curve (20). This gives the 'speed' of any 
free mode in terms of that of the corresponding mode when there is no rotation. 

211. As a sufficient example oi forced oscillations we may assume 

f = Q' e<<'*+'«+'», (23) 

where the value of a is now prescribed. 

This makes V^'f "^ 0> ^^^ ^^^ equation (4) then gives 

J = AJ, {kt) e<«^«+'«+'), (24)' 

where il is to be determined by the boundary-condition (7), viz. 

2a> 



8 



('-t) 



A i ^ .0 (25) 

o • 

This becomes very great when the frequency of the disturbance is nearly 
coincident with that of a free mode of corresponding type*. 

From the point of view of tidal theory the most interesting cases ore those of « = 1 
with cr=oi>, and ^=3 2 with <r=2<0, respectively. These would represent the diurnal 
and semidiurnal tides due to a distant disturbing body whose proper motion may be 
nciglected in comparison with the rotation ». 

In the case of « = ! we have a ttm/ofm horizontal disturbing force. Putting, in 
addition, o- = <», we find without difficulty that the amplitude of the tide-elevation at the 
edge (r =a) of the basin has to its * equilibrium- value' the ratio 

A(z)+2/o(2) ^^^^ 

where z^\ n/(3/3). With the help of Lodge's tables we find that this ratio has the values 

1-000, -638, -396, 

for j3=: 0, 12, 48, respectively? 

When (T =2o>, we have k =0, and thence, by (23), (24), (25), 

f=f, (27) 

i.e. the tidal elevation has exactly the equilibrium- value. 

This remarkable result can be obtained in a more general manner; it holds whenever 
the disturbing force is of the type 

f=;^(r)e««»*+^+*) (28) 

provided the depth A be a function of r only. If we revert to the equations (1)» we notice 

* The case of a neaWy circular sheet is treated by Proudman, " On some Cases of Tidal Motion 
on Rotating Sheets of Water," Froc. Lond. Maih. 8oc. t. xii. p. 463 (1913). 



316 Tidal Waves [chap, vin 

that when cr=2tt they are satisfied by C=C* O^^-S* To determine i2 as a function 
of r, we substitute in the equation of continuity (2), which gives 



• •«•■ ■ * ••■• 



• ^^-t^l^^-^^r) (29) 

The arbitrary constant which appears on integration of this equation is to be determined 
by the boundary-condition. 

In the present case we have x(^)=^^/a'* Integrating, and making 12=0 -for r=a, 
we find 

^ij=^'(a«-r«) «'<•-*+••+•> (30) 

The relation Q=%R shews that the amplitudes of R and e are equal, while their phases 
differ by 00° ; the relative orbits of the fluid particles are in fact circles of radii 

'=2*5^(»'-'*) <»1) 

described each about its centre with angular velocity 2» in the negative direction. We 
may easily deduce that the path of any particle in space is an ellipse of semi-axes r±T 
described about the origin with harmonic motion in the positive direction, the period 
being 2irla.^ This accounts for the peculiar features of the case. For if ( have always the 
equilibrium- value, the horizontal forces due to the elevation exactly balance the disturbing 
force, and there remain only the forces due to the undisturbed form of the free surface 
(Art. 207 (1)). These give an acceleration gdzjdr, or <h>V, to the centre, where r is 
the jradius vector of the particle in its actual position. Hence all the conditions of the 
problem are satisfied by elliptic-harmonic motion of the individual particles, provided the 
positions, the dimensions, and the 'epochs' of the orbits can be adjusted so as to satisfy 
the condition of continuity, with the assumed value of (» The investigation just given 
resolves this point. 

212*. We may also briefly notice the case of a circular basin of variable 
depth, the law of depth being the same as in Art. 193, viz. 



* = *o(l-J.) (1) 



Assuming that i^, 6, ^ all vary as e'*'*+**+*^ and that A is a function of r only, 
we find, from Art 209 (2), (3), 

Introducing the value of h from (1), we have, for theJVee oscillations, 

Tins is identical with Art. 193 (6), except that we now have 

ghQ aa* 

in place of cr^/gh^. The solution can therefore be written down from the results of that 
Art., viz. if we put 

__ _=n(n-2)-^ (4) 

* See the footnote to Art. 193. 



211-212] Badn of Variable Depth 317 

wehave C =4. Q' J (a, A y. ^,) e* "*+--^-' (6) 

where 'a='|nH^J4, j8 = l+J«— J«, ystf + l; 

and the condition of oonveigence at the boundary r =a requires that 

»=»+2j (6) 

where j is some poflitive integer. The values of o- are then given by (4). 

The forms of the free surface are therefore the same as in the ease of no rotation, but 
the motion of the water-particles is different. The relative orbits are in fact now ellipses 
having their principal axes along and perpendicular to the radius vector; this follows 
easily from Art. 209 (3). 

In the symmetrieal modes (a =0), the equation (4) gives 

cr« = <ro« + 4««, (7) 

where o-q denotes the 'speed' of the corresponding mode in the case of no rotation, as 
found in Art. 193. 

For any value of a other than zero, the most important modes are those for which 
n=« + 2. The equation (4) is then divisible by cr+2<io, but this is an extraneous factor; 
discarding it, we have the quadratic 

«r«-2fi>(r=2«?^°, (8) 



(9) 



whence o- =» ± f «* +2« -^ j 

This gives two waves rotating round the origin, the relative wave-velocity being greater 
for the negative than for the positive wave, as in the case of uniform depth (Ait. 210). 
With the help of (8) the formulae reduce to 

f=^.(9'. ^=^i^.($j'\ «=*»£^.©'" <»«) 

the factor e' '''*'*"**'*^^ being of course understood in each case. Since e=ti?, the relative 
orbits are all circles. The case « = 1 is noteworthy; the free surface is then always plane, 
and the circular orbits have all the same radius. 

When n>« +2, we have nodal circles. The equation (4) is then a cubic in 0-/20); it is 
easily seen that its roots are all real, lying between — oo and —1, —1 and 0, and +1 
and + 00 , respectively. As a numerical example, in the case of « = 1, n =6, corresponding 
to the values 

2, 6, 40 

of ^*a^lg\, we find 

( + 2-889 + 1-874 + M80, 

- 0-125 -0-100 -0-037, 

-2-764 -1-774 -1-143. 

The first and the last root of each triad give positive and negative waves of a somewhat 
similar character to those already obtained in the case of uniform depth. The smaller 
negative root gives a comparatively slow oscillation which, when the angular velocity o> is 
infinitely small, becomes a steady rotational motion, without elevation or depression of the 
surface*. 

* The possibility of oscillations of this type was pointed out in Art. 206, ad fin. 



318 Tidal Waves [chap, vni 

The most important type oi forced oscillations is such that 

f=0gYe<<'*+'*-^> (11) 

We readily verify, on substitution in (3), that 

'"'2«^Ao~(cr*-2«ir)a«^ ^^^^ 

We notice that when crs2«> the tide-height has exactly the equilibrium-value, in agree- 
ment with Art. 211. 

If (Ti, cTi denote the two roots of (8), the last formula may be written 

^"(l-cr/crjd-cr/cr,) <^^) 

Tides on a Rotating Globe. 

213. We proceed to give some account of Laplace's problem of the tidal 
oscillations of an ocean of (comparatively) small depth covering a rotating 
globe*. In order to bring out more clearly the nature of the approximations 
which are made on various groimds, we adopt a method of establishing the 
fundamental equations somewhat different from that usually followed. 

When in relative equilibrium, the free surface is of course a level-surface 
with respect to gravity and centrifugal force; we shall assume it to be 
a surface of revolution afcout the polar axis, but the ellipticity will not in 
the first instance be taken to be small. 

We adopt this equilibrium-form of the free surface as a surface of 
reference, and denote by 6 and <f> the co-latitude (i.e. the angle which the 
normal makes with the polar axis) and the longitude, respectively, of any 
point upon it. We shall further denote by z the altitude, measured outwards 
along a normal, of any point above this surface. 

The relative position of any particle of the fluid being specified by 
the three orthogonal co-ordinates 0, <^, z, the kinetic energy of unit mass 
is given by 

2T^(R-{-zYd^ + tn^{oy^4>Y-\-z\ (1) 

where 2?' is the radius of curvature of the meridian-section of the surface of 
reference, and m is the distance of the particle from the polar axis. It is to 
be noticed that i2 is a function of d only, whilst to is a function of both 6 and 
z ; and it easily follows from geometrical considerations that 

= cos ff, -5- = sm (2) 



(R-\-z)dd ' dz 

* "Recherches sur quelquee points du syst^me du monde," Mim, de VAcad, toy, de» Sciences, 
1775 [1778] and 1776 [1779]; Oeuvres Completes, t. ix. pp. 88, 187. The mvestigation is repro- 
duced, with various modifications, in the Micanique Celeste, livre 4"**, c. i. (1799). 



212-213] Tides on a Rotating Globe 319 

The component accelerations are obtained at once from (1) bj Lagrange's 
formula. Omitting terms of the second order, on account of the restriction 
to infinitelj small motions, we have 

1 fddT dT\ ,D^ .x 1 / , lo ix ^o* 



1 fd dT dT\ 1 ^ ^ f^m A , dm A 



\ ..(3) 



dt dz dz dz' 

Hence, if we write u, v, w for the component relative velocities of a 

particle, viz. 

U'={R + z)0, v = w^, w^z, (4) 

and make use of (2), the hydrodynamical equations may be put in the forms 



|^-2a«;cose . . ^^| (? + ^ . J^.^a + q)/ 



5^4- 2ci>ttcos&+ 2a>wsine= - ~ ^ f- + ^ - W^^ + Q), 



I., (5) 



^-2a>t;sinfl = ~ ^ (? + ^ - Jco«m« + q), 

where ^ is the gravitation-potential due to the earth's attraction, whilst £2 
denotes the potential of the disturbing forces! 

So far the only approximation has consisted in the omission of terms of 
the second order in ti, t;, w. In the present application, the depth of the sea 
being small compared with the dimensions of the globe, we may replace 
R-\- z by fi. We will further assume that the vertical velocity w is small 
compared with the horizontal components u, v, and that dwjdt may be 
neglected in comparison with oyv. As in the theory of 'long' waves, 
such assumptions are justified a posteriori if the results obtained are 
found to be consistent with them (cf. Art. 172). 

Let us integrate the third of equations (5) between the limits z and £, 
where J denotes the elevation of the disturbed surface above the surface of 
reference. At the surface of reference (z «= 0) we have 

^ — Jco^tu* = const., 

by hypothesis, and therefore at the free surface {z^ t) 

\ff ^ ^w^xn^ = const. + ^{, 



(6) 



approximately, provided ^ = g- (^ — Jcu^id*) 

Here g denotes the value of apparent gravity at the surface of reference; 
it is of course, in general, a function of 0, but its variation with z is 
neglected. 



320 Tidal Waves [Chap, vin 

*The integration in question then gives 

2 + >I^ - \a>^w^ = const. + ^f + 2a) sin & | vdz, (7) 

P J e 

where the variation of the disturbing potential Q with z has been neglected 
in comparison with g. The last term is of the order of whv sin d, where h is the 
depth of the fluid, and it may be shewn that in the subsequent applications 
this is of the order A/a as compared with g^. Hence, substituting in the first 
two of equations (5), we obtain, with the approximations indicated, 

^-2««;co80=-A<,(^-r). | + 2«,«cose=-^^(S-f),..(8) 

where f=-£i/^ (9) 

These equations are independent of z, so that the horizontal motiou 
may be assumed to be sensibly the same for all particles in the same vertical 
line. 

As in Art. 198, this last result greatly simplifies the equation of continuity. 
In the present case we find without difficulty 

d^_ i( d jhwu) d (hv)] 

dt~ m\ Rde "^ d<l> ] ^ ^ 

It is important to notice that the preceding equations involve no 
assumptions beyond those expressly laid down; in particular, there is no 
restriction as to the ellipticity of the meridian, which may be of any degree 
of oblateness. 

214. In order, however, to simplify the question as far as possible, 
without sacrificing any of its essential features, we now take advantage 
of the circumstance that in the actual case of the earth the ellipticity is 
a small quantity, being in fact comparable with the ratio {lo^ajg) of centrifugal 
force to gravity at the equator, which ratio is known to be about ^. Subject 
to an error of this order of magm'tude, we may put fi = a, id = a sin fl, 
g = const., where a is the earth's mean radius. We thus obtain 

|-2a..cos^=-f|(C--r), |+2a>«COB^=-f^^(J-a 

(1) 



with 



dt^ 1 ■ 

dt a sin d 



d (hu sin d) d (hv) 



+ ^- , (2) 



dd ' d<i> 

this last equation being identical with Art, 198 (1)*. 

Some conclusions of interest foUow at once from the mere form of the 
equations (1). In the first place, if u, v denote the velocities along and 

* Except for the notation these are the equations arrived at by Laplace, tc. ante p. 318. 



213-215] Tides on a Rotating Globe 321 

perpendicular to any horizontal direction 8, we easily find, by transforma- 
tion of co-ordinates 

|^-2a,vco8d=-<7|(C-?) (3) 

In the case of a narrow canal, the transverse velocity v is zero, and the 
equation (3) takes the same form as in the case of no rotation ; this has been 
assumed by anticipation in Art. 183. The only effect of the rotation in such 
cases is to produce a slight slope of the wave-crests and furrows in the 
direction across the canal, as investigated in Art. 208. In the general case, 
resolving at right angles to the direction of the relative velocity {q, say), we 
see that a fluid particle has an apparent acceleration 2o)q cos d towards the 
right of its path, in addition to that due to the forces. 

Again, by comparison of (1) with Art. 207 (5), we see that the oscillations 
of a sheet of water of relatively small dimensions, in colatitude d, will take 
place according to the same laws as those of a plane sheet rotating about 
a normal to its plane with angular velocity ay cos d. 

As in Art. 207, free steady motions are possible, subject to certain 
conditions. Putting ^=0, we find that the equations (1) and (2) are 
satisfied by constant values of w, v, J, provided 

u_ ^ ^^ v^ ^ ^^ (4) 

2a>a sin d cos 6 d<f) ' 2a>a cos Odd^ 

'-^^mw-" «>' 

The latter condition is satisfied by any assumption .of the form 

S=/(Asecfl) (6) 

and the equations (4) then give the values of u, v. It appears from (4) that 
the velocity in these steady motions is everywhere parallel to the contour- 
lines of the disturbed surface. 

If h is constant, or a function of the latitude only, the only condition 
imposed on t, is that it should be independent of <f)\ in other words the 
elevation must be symmetrical about the polar axis. 

215. We shall suppose henceforward that the depth A is a function of B 
only, and that the barriers to the sea, if any, coincide with parallels of 
latitude. 

We take first the cases where the disturbed form of the water-surface 
is one of revolution about the polar axis. When the terms involving <f> are 
omitted, the equations (1) and (2) of the preceding Art. take the forms 

^ - 2o)V cos = - - ^ (^ - f), X- + %JDU cos & = 0, (1) 

01 a ou ut 

with i=-— -t.? (2) 

at a sm Odd 

L. H. 21 



322 Tidal Waves [chap, vin 

Aflsuming a time-factor 6^^ and solving for u, v, we find 

a« - 4a>« cos« 0add^^ ^^' ^ " (t« - 4coa cos* dadd^^" ^^' " ' ^^ 

.., , y d (Aw sin 0) ... 

with taj = - — .-- o^ (4) 

The formulae for the component displacements (f, iy, say) can be written 
down from the relations w = f , t; = ^, or w = taj, v = icny. It appears that the 
fluid particles describe ellipses having their principal axes along the meridians 
and the parallels of latitude, respectively, the ratio of the axes being 
a/2a} . sec 6, In the forced oscillations of the present type the ratio a/2o} is 
very small ; so that the ellipses are very elongated, with the greatest length 
from E. to W., except in the neighbourhood of the equator. 

Eliminating u and t; between (3) and (4), and writing, for shortness, 
In the case of uniform depth, this becomes 

|;(^.|)h-«'-« (') 

where /a = cos 0, and j3 = -r— = — , - (8) 

216. First, as regards the/re6 oscillations. Putting f = 0, we have 

I; (t^. a^) + « - ». <') 

and we notice that in the case of no rotation this is included in (1) of Art. 199, 
as may be seen by putting Pf^ = (J^a^/gh,f = oo . The general solution of (9) 
is necessarily of the form 

C = ^l* {/*) + 5/ (m), (10) 

where F {fi) is an even, and / (fi) an odd, function of ^, and the constants 
A, B are arbitrary. In the case of a zonal sea bounded by two parallels of 
latitude, the ratio A : B, and the admissible values of / (and thence of the 
frequency a/2n) are determined by the conditions that m = at each of these 
parallels. If the boundaries are symmetrically situated on opposite sides 
of the equator, the oscillations fall into two classes; viz. in one of these 
B = 0, and in the other ^ = 0. By supposing the boundaries to contract to 
points at the poles, we pass to the case of an unlimited ocean, and the 
admissible values of / are now determined by the condition that u must be 
finite f or ^ = ± 1. The argument is, in principle, exactly that of Art. 201, 



215-216] Case of Symmetry 323 

but the application of the last-mentioned condition is now more difficult, 
owing to the less familiar form in which the solution of the differential 
equation is obtained. 

In the case of symmetry with respect to the equator, we assume, 
following the method of Kelvin* and Darwin t, 

^^^4r^|^ = Bi^ + B3M»+...+Ba/+i/^«+^ (11) 

This leads to 

(12) 

where A is arbitrary ; and makes 

(13) 

Substituting in (9), and equating coefficients of the several powers of /t, 
we find 

B^-^A = (14) 

^3 - (l - If^) 5i = 0, (15) 

and thenceforward 

^2^+1 - (l - 2i (2] + 1)) ^*^-^ " ^W+^ ^^'^ " ^ ^^^^ 

These equations determine B^^B^, ... B^s^^y ... in succession, in terms of 
A, and the solution thus obtained would be appropriate, as already explained, 
to the case of a zonal sea bounded by two parallels in equal N. and S. latitudes. 
In the case of an ocean covering the globe, it would, as we shall prove, give 
infinite velocities at the poles, except for certain definite values of/. 

Let us write ^ay+i/^a^-i = -^i+iJ (17) 

we shall shew, in the first place, that as j increases N^ must tend either to 
the limit or to the limit 1. The equation (16) may be written 

"^'^^ 23{23-hiy2j(23 + \)N, ^^^^ 

Hence, when j is large, either 

^*^~ 2j"(2; + l)' (^^) 

* Sir W. Thomson, "Note on the 'Oscillations of the First Species' in Laplace's Theory of 
the Tides," PhU, Mag. (4), t. L p. 279 (1876) [Papers, t. iv. p. 248]. 

t "On the Dynamical Theory of the Tides of Long Period," Proe. Roy, 8oc, t. xlL p. 337 
(1886) [Papers, t. i. p. 366], 

21—2 



324 Tidal Waves [ohap. vm 

approximately, or N^^^ is not small, in which case N^^^ ^^ ^^ nearly equal 
to 1, and the values of iV^+8> ^i+4> • • • ^^ ^^^^ more and more nearly to 1, 
the approximate formula being 

^^+^=^~2i(2i+l) ^^^^ 

Hence, with increasing y, N^ tends to one or other of the forms (19) and (20). 

In the former case (19), the series (11) will be convergent for ^ = ± 1, and 
the solution will be valid over the whole globe. 

In the other event (20), the product N^N^ ... JVy+i, and therefore the. 
coefficient 52/+i> tends with increasing ^ to a finite limit other than zero. 
The series (11) will then, after some finite number of terms, become com- 
parable with 1 4- /A* + ^* 4- . . . , or (1 — /x*)~^, so that we mav write 

■ ' «'-i+.^ ' m 



where L and M are functions of /a which remain finite when /a »= ± 1. Hence, 
from (3), 

»=-£7^* I'— £'"-'->*^+(>-''*)"' ■«>•■•" 

which makes u infinite at the poles. 

It follows that the conditions of our problem can be satisfied only if Nf 
tends to the limit zero ; and this consideration, as we shall see, restricts us to 
a determinate series of values of/. 

The relation (18) may be put in the form 

ff, "W^ (23) 

1 PJ _ AT 

and by successive applications of this we obtain N^ in the form of a 
convergent continued fraction 

j3 _ _p ^ 

~2i(2i+l) (2j+ 2yW+"3) (2.7 + 4) {2j + 5) 

PP ] W \ j3/' 

^~2/(2j>I)"^ ^■"(2j+2)(2j + 3)+ ^~(2i+4)(2; + 5)"^-" 

(24) 

on the present supposition that Nj^j^ tends with increasing k to the limit 0, 
in the manner indicated by (19). In particular, this formula (24) determines 
the value of JVj. Now from (15) we must have 

iV,= l-|4' (25) 



216-217] Free OsdUations 326 

which is equivalent to iV^ = oo . This equation determines the admissible 
values of /{= (T/2a)). The constants in (11) are then given by 

Bi = pA, Ba = N^PA, B^ = N^N^PA, . . . , .... (27) 

where A is arbitrary. 

It is easily seen that when j3 is infinitesimal the roots of (26) are given by 



a^a^ 



= i3/« = w(n+l), (28) 



gh 
where n is an even integer ; cf . Art. 199. 

One arithmetically remarkable point remains to be noticed. It might 
appear at first sight that when a value of / has been found from (26) the 
coefficients B^, B^, B^y ... could be found in succession from (15) and (16), or 
by means of the equivalent formula (18). But this would require us to start 
with exactly the right value of / and to observe absolute accuracy in the 
subsequent stages of the work. The above argument shews, in fact, that any 
other value, difiering by however little, if adopted as a starting point for the 
calculation will inevitably lead at length to values of Nf which approximate 
to the limit 1*. 

217. It is shewn in the Appendix to this Chapter that the tide- 
generating potential, when expanded in simple-harmonic functions of the 
time, consists of terms of three distinct types. 

The first type is such that the equilibrium tide-height would be given by 

f = JBT ' (i - COS* 0) . cos ((T« -F €) t (29) 

The corresponding forced waves are called by Laplace the ' Oscillations of the 
First Species' ; they include the lunar fortnightly and the solar semi-annual 
tides, and generally all the tides of long period. Their characteristic is 
symmetry about the polar axis, and they form accordingly the most important 
case of forced oscillations of the present type. 

If we substitute from (29) in (7), and assume for 



1^* ^' and U 



* Sir W. Thomson, le, ante p. 323. 

t In strictnefis, 6 heie denotes the geocentric latitude, but the difference between this and the 
geographical latitude may be neglected consistently with the assumptions introduced in Art. 214. 



326 Tidal Waves [chap, vm 

expressions of the forms (11) and (12), we have, in place of (14), (15), 

5, - i/S^' - /3^ = (30) 

/3/« 



5.-(l-|^)5i + ii8£r' = 0, (31) 



whilst (16) and its consequences hold for all the higher coefficients. It may 
be noticed that (31) may be included under the general formula (16), provided 
we write B_^ = — 2H\ It appears by the same argument as before that the 
only admissible solution for an ocean covering the globe is the one that makes 
N^ = 0, and that accordingly iV^^ must have the value given by the continued 
fraction in (24), where /is now prescribed by the frequency of the disturbing 
forces. 

In particular, this formula determines the value of N^. Now 

and the equation (30) then gives 

A^-^H'-^N^W; (32) 

in other words, this is the only value of A which. is consistent with a zero 
limit of Nj, and therefore with a finite velocity at the poles. Any other value 
of A, if adopted as a starting point for the calculation of jB^, £3, £5, ... in 
succession, by means of (30), (31), and (16), would lead ultimately to values 
of Ni approximating to the limit 1. Moreover, since abaoVute accuracy in the 
initial choice of A and in the subsequent computations would be essential to 
avoid this, the only practical method of calculating the coefficients is to use 
the formulae 

B^H' = - 2iVi, £3 = N^Bj^, B, = N^B^, . . . , 

or BJH' = -2N^, B^jH' = -2N^N^, BJH' ^ ^2N^N^N^, .. . 

(33) 

where the values oi Ni, N^, N^y ... are to be computed from the continued 
fraction (24). It is evident a posteriori that the solution thus obtained will 
satisfy aH the conditions of the problem, and that the series (12) will converge 
with great rapidity. The most convenient plan of conducting the ctdculation 
is to assume a roughly approximate value, suggested by (19), for one of the 
ratios N^ of sufficiently high order, and thence to compute 

in succession by means of the formula (23). The values of the constants 
AjBiyB^, . . . , in (12), are then given by (32) and (33). For the tidal elevation 
we find 

^/H' = - 2NJ^ - (1 -f*N,) ,*» - iN, (1 -/W.) /*«-... 

- -. NiNt . . . Ni_i (1 -f*Ni) /*« - (34) 



217] Tides of Long Period 327 

In the case of the lunar fortnightly tide, / is the ratio of a sidereal day 
to a lunar month, and is therefore equal to about ^, or more precisely '0365. 
This makes/* = '00133. It is evident that a fairly accurate representation 
of this tide, and d, fortiori of the solar semi-annu^ tide, and of the remaining 
tides of long period, will be obtained by putting / = ; this materially 
shortens the calculations. 

The results will involve the value of j8, = 4ai*a*/^A. For j8 = 40, which 
corresponds to a depth of 7260 feet, we find in this way 

ijE' = -1515 - l'0000/i« + 1'5153/A* - l'2120/i« + '6063/^8 - •2076/ii<> 

+ -0516/^" - 0097/1" + OOlS/Lii* - 0002/1", (35)* 

whence, at the poles (/a = ± 1), 

i = - f ff ' X '154, 

and, at the equator (/i = 0), 

S = iff ' X -455. 

Again, for jS = 10, or a depth of 29040 feet, we get 

IjH' = '2359 - lOOOO/i* + -5898/1* - '1623/i« 

+ -0258/18 - 0026/110 -f -0002/112 (36) 

This makes, at the poles, 

^=-|ff' X '470, 

and, at the equator, 

• {= iff ' X '708. 

For jS = 5, or a depth of 58080 feet, we find 
IjH' = -2723 - lOOOO/i* + '3404/i* 

- '0509/i« + '0043/i8 - -0004/iio (37) 

This gives, at the poles, 

S = - Iff ' X -651, 

and, at the equator, 

i = iff ' X '817. 

Since the polar and equatorial values of the equilibrium tide are — fff ' 
and iff', respectively, these results shew that for the depths in question 
the long-period tides are, on the whole, direct, though the nodal circles will, 
of course, be shifted more or less from the positions assigned by the equi- 
librium theory. It appears, moreover, that, for depths comparable with the 
actual depth of the sea, the tide has less than half the equilibrium value. 
It is easily seen from the form of equation (7) that with increasing depth, 
and consequent diminution of j3, the tide-height will approximate more and 
more closely to the equilibrium value. This tendency is illustrated by the 
above numerical results. 

* The ooeffioiente in (35) and (36) differ only slightly from the numerical values obtained by 
Darwin for the case /= '0365. 



328 



Tidal Waves 



[CHAP, vm 



It is to be remarked that the kinetic theory of the long-period tides was 
passed over by Laplace, under the impression that practically, owing to the 
operation of dissipative forces, they would have the vtdues given by the 
equilibrium theory. He proved, indeed, that the tendency of frictional forces 
must be in this direction, but it has been pointed out by Darwin* that in 
the case of the fortnightly tide, at all events, it is doubtful whether the effect 
would be nearly so great as Laplace supposed. We shall return to this point 
later. 

218. When the disturbance is no longer restricted to be symmetrical 
about the polar axis, we must recur to the general equations (1) and (2) of 
Art. 214. We retain, however, the assumptions as to the law of depth and 
the nature of the boundaries introduced in Art. 215. 

If we assume that i2, w, v, { all vary as e<f<'*+^+«), where $ is integral, the 
equations referred to give 



icFU — %A)V COS fl = — - ;^ ({ — ?), icFV -\- 2cott cos fl = — . - 



....(1) 



with 

Solving for w, v, we find 



. ^ _ 1 (3 (hu sin B) , . 



-fwAvl (2) 



4m (/2 — cos^ 



a /cosfl3J' . w f\ 



(3) 



where we have written 
as before. 



2a> 



=/. 



o}^a 



= m, 



(4) 



It appears that in all cases of simple-harmonic oscillation the fluid 
particles describe ellipses having their principal axes along the meridians 
and parallels of latitude, respectively. 

Substituting from (3) in (2) we obtain the differential equation in J' : 

_ ., ^ .fl flcot e^^ + «»r cosec* e) + ifmi' = - ima^. . . (6) 
/* — cos* d\f 08 I 



* l.c. anU p. 323. 



217-219] Diurnal Tides 329 

219. The case « » 1 includes, as forced oscillations, Laplace's ' Oscillations 
of the Second Species,' where the disturbing potential is a tesseral harmonic 
of the second order ; viz. 

f «- ff " sin fl cos fl . cos (a« H- ^ + €), (1) 

where a differs not very greatly from w. This includes the limar and solar 
diurnal tides. 

In the case of a disturbing body whose proper motion could be neglected, 
we should have a = a>, exactly, and therefore/ = J- In the case of the moon, 
the orbital motion is so rapid that the actual period of the principal lunar 
diurnal tide is very appreciably longer than a sidereal day*; but the sup- 
position that/= ^ simplifies the formulae so materiaUy that we adopt it in 
the following investigation f. We find that it enables us to ccJculate the 
forced oscillations when the depth follows the law 

A = (1 - gr COS* fl) Ao, (2) 

where q is any given constant. 

Taking an exponential factor 6*<'**+*+*\ and therefore putting * = 1,/= i, 
in Art. 218 (3), and assuming 

r=Csinflcosfl, (3) 

C C 

we find w=— t<7— , v = a— . cos (4) 

mm 

Substituting in the equation of continuity (Art. 218 (2)), we get 

£'+?-^S (^' 

which is consistent with the law of depth (2), provided 

^==~l-2?Ao/ma^" ^^^ 

Thisgives ^^_ 2gVma ^ 

^ ^ 1 - 2qhQ/ma * ^ ' 

One remarkable consequence of this formula is that in the case of uniform 
depth {q = 0) there is no diurnal tide, so far as the rise and fall of the surface 
is concerned. This result was first estabb'shed (in a different manner) by 
Laplace, who attached great importance to it as shewing that his kinetic 
theory was able to account for the relatively small values of the diurnal tide 

* It is to be remarked, however, that there is an important term in the harmonic develop- 
ment of for which <r = ta exactly, provided we neglect the changes in the plane of the disturbing 
body's orbit. This period is the same for the sun as for the moon, and the two partial tides thus 
produced combine into what is caUed the *luni-solar* diurnal tide. 

t Taken with very slight alteration from Airy, "Tides and Waves," Arts. 96 ... , and 
Darwin, Eneye, BriU 9th ed., t. xxiii. p. 359. 



330 Tidal Waves [chap, vm 

which are given by observation, in striking contrast to what would be 
demanded by the equilibrium theory. 

But, although with a uniform depth there is no rise and fall, there are 
tidal currents. It appears from (4) that every particle describes an ellipse 
whose major axis is in the direction of the meridian, and of the same length 
in all latitudes. The ratio of the minor to the major axis is cos0, and so 
varies from 1 at the poles to at the equator, where the motion is wholly 
N. and S. 

220. In* the case a = 2, the forced oscillations of most importance are 
where the disturbing potential is a sectorial harmonic of the second order. 
These constitute Laplace's * Oscillations of the Third Species,' for which 

f = H'" sin«fl . cos (a« + 2^ + €), (1) 

where a is nearly equal to 2a>. This includes the most important of all the 
tidal oscillations, viz. the lunar and solar semi-diurnal tides. 

If the orbital motion of the disturbing body were infinitely slow we should 
have a = 2co, and therefore/ = 1 ; for simplicity we follow Laplace in making 
this approximation, although it is a somewhat rough one in the case of the 
principal lunar tide*. 

A solution similar to that of the preceding Art. can be obtained for the 
special law of depth f 

A = Aosin*fl (2) 

Adopting an exponential factor e*l*"*+**+*^ and putting therefore/ = 1, « = 2,. 
we find that if we assume 

r = Csin^fi, (3) 

the equations (3) of Art. 218 give 

u = — C cot 6, t?=-s-C,- r-^, (4) 

m 2m 1 + cos^fi ^ ' 

whence, substituting in Art. 218 (2), 

i = ?^.Csin«fi (5) 

ma ^ ' 

Putting C = r + t> ^^^ substituting from (1) and (3), we find 

C^-', ^-r-H"\ (6) 

1 — 2hQ/ma ^ ' 

and therefore C=- . ^V/T I (7) 

* There is, however, a *lniu-8olar' semi-diomal tide whose speed is exactly 2w if we neglect 
the changes in the planes of the orbits. Cf. p. 329, first footnote. 
t Cf. Airy and Darwin« U.ee, 



219-221] Semi-diurnal Tides 331 

For such depths as actually occur in. the ocean 2Ao < ma, and the tide is 
therefore inverted. It may be noticed that the formulae (4) make the velocity 
infinite at the poles, as was to be expected, since the depth there is zero. 

221. For any other law of depth a solution can only be obtained in the 
form of a series. In the case of uniform depth we find, putting * = 2,/= 1, 
ima/h = j3 in Art. 218 (5), 

^^ ~ ^'^'l^ + {^(1 - H'')' - 2,.« - 6} r = - i8(l - firl ..(8) 

where /i is written for cos 0. In this form the equation is somewhat intract- 
able, since it contains terms of four different dimensions in /i. It simplifies 
a little, however, if we transform to 

V, =(1 -,.«)*, =sinfl, 
as independent variable ; viz. we find 

^i^-^) ^5' -v^-{8-2v»- /3v«) ^' = - j8,^f = - ^H'" A . .(9) 

which is of three different dimensions in v. 

To obtain a solution for the case of an ocean covering the globe, we assume 

$' = Bo + B^v* + B^iA + . . . + B^v^ + (10) 

Substituting in (9), and equating coeflScients, we find 

Bo = 0, B, = 0, 0.^4 = 0, (11) 

16Be - IOB4 + j8ff '" = 0, (12) 

and thenceforward 

■ 

2; (2j + 6) B^^ - 2j {2j + 3) 5«+, + )8B^ = (13) 

These equations give B^, B^, ... B^j, ... in succession, in terms of B^, which 
is so far undetermined. It is obvious, however, from the nature of the 
problem, that, except for certain special values of h (and therefore of j8), 
which are such that there is a free oscillation of corresponding type {s = 2) 
having the speed 2co, the solution must be unique. We shall see, in fact, 
that unless B^ have a certain definite value the solution above indicated will 
make the meridian component (u) of the velocity discontinuous at the 
equator*. 

The argument is in some respects similar to that of Art. 217. If we 
denote by N^ the ratio B^j+JBii of consecutive coefficients, we have, from (13), 

2i + 3 P 1 

2j + 6 2j(2j-\-6)N,' 



i^..x = ^4^-Kr7.?^.i^, (14) 



< 



* In the case of a polar sea bounded by a smaU circle of latitude whose angular radius is 
ir, the value of B4 is determined by the condition that u^O,or d^/dif =0, at the boundary. 



332 Tidal Waves [chap, vm 

from which it appears that, with increasing j, Nj muBt tend to one or other 
of the limits and 1. More precisely, unless the limit of N^ be zero, the 
limiting form of iV^+i will be 

(2i + 3)/(2i + 6), or 1 - |, 

approximately. The latter is identical with the limiting form of the ratio 
of the coefficients of v^ and v^~* in the expansion of (1 — v^y. We infer that, 
unless £4 have such a value as to make N^ = 0, the terms of the series (10) 
will become ultimately comparable with those of (1 — v*)*, so that we may 
write 

r = i+(l-v«)*M, (15) 

where i, M are functions of v which do not vanish for v = 1. Near the 
equator (v = 1) this makes 

S=^(i-'*)*f=±^ (i«) 

Hence, by Art. 218 (3), u would change from a certain finite value to an 
equal but opposite value as we cross the equator. 

It is therefore essential, for our presept purpose, to choose the value of B^ 
so that N^ = 0. This is effected by the same method as in Art. 217. Writing 
(13) in the form 

N.-MS^ (") 

2j + 6 ^'« 

we see that N^ must be given by the converging continued fraction 

j3 )3 j3 

2i(2i+6)(27+2)(2i + 8) (^j + 4) (2i + 10) 

2j + 3 2j + 5 2j + 7 ^ ^^^^ 

2^ + 6 2j + 8 2i+10 *''• 

This holds from J = 2 upwards, but it appears from (12) that it will give also 
the value of N^ (not hitherto defined), provided we use this symbol for BJH"\ 
We have then 

B^ = N^H''\ B, = N,B^, Bs = ^s^e, .... 

Finally, writing C = f + T* we obtain 

^/H'" = v« + N^i^ + N^Ny + N^N^N^i/^ + (19) 

As in Art. 217, the practical method of conducting the calculation is to 
assume an approximate value for iV^+i, where j* is a moderately large number, 
and then to deduce Nj, ^y-i, ... N^, N^ in succession by means of the 
formula (17). 



221] 



Semi'diurnal Tides 333 



The above investigation is taken substantially from the very remarkable paper written 
by Kelvin* in vindication of Laplace's treatment of the problem, as given in the 
MAanique Cdeste. In the passage more especially in question, Laplace determines the 
constant B^ by means of the continued fraction for Ni, without, it must be allowed, 
giving any adequate justification of the step; and the soundness of this procedure 
had been disputed by Airyf, and after him by Ferrel}. 

Laplace, unfortunately, was not in the habit of giving specific references, so that few of 
his readers appear to have become acquainted with the original presentment! of the 
kinetic theory, where the solution for the case in question is put in a very convincing, 
though somewhat different, form. Aiming in the first instance at an approximate 
solution by means of a finite series, thus : 

C=B^v* +Bf^u* + . . . + Btt + 2»'****. (20) 

Laplace remarks || that in ojder to satisfy the differential equation, the coefficients would 
have to fulfil the conditions 

16Be-10JB4+/3JI'"=0, 

40JB8 - 28Be + /3B4 = 0, 



y (21) 



(2fc -2) (2ifc+4)Btt+, -(2ifc -2) (2ifc + l)Btt+/3B,jt.8=0, 

-2fc (2ifc +3) Btk+2 +^Btk =0, 

^-82*+ 2 =0,-^ 

as is seen at once by putting Buc^^ =0, Bfjc^t=0, ... in the general relation (13). 

We have here k + l equations between k constants. The method followed is to 
determine the constants by means of the first k relations; we thus obtain an exact 
solution, not of the proposed differential equation (9), but of the equation as modified by 
the addition of a term /SBu+tv^'^* to the right-hand side. This is equivalent to an 
alteration of the disturbing force, and if we can obtain a solution such that the required 
alteration is very small, we may accept it as an approximate solution of the problem 
in its original form^. 

Now, taking the first k relations of the system (21) in reverse order, we obtain B^k^t 
in terms of B^k, thence B^k in terms of B^^i, and so on, until, finally, B4 is expressed in 
terms of H"'; and it is obvious that if A; be large enough the value of B2k+t, and the 
consequent adjustment of the disturbing force which is required to make the solution 
exact, will be very smalL This will be illustrated presently, after Laplace, by a numerical 
example. 

The process just given is plainly equivalent to the use of the continued fraction (17) 
in the manner already explained, starting with J + 1=A;, and Nk=fi/2k{2k+S). The 
continued fraction, as such, does not, however, make its appearance in the memoir here 
referred to, but was introduced in the MScanique Cdeste, probably as an after-thought, as a 
condensed expression of the method of computation originally employed. 

♦ Sir W. Thomaon, "On an Alleged Error in Laplace's Theory of the Tides," Phil. Mag. 
r4), 1. 1. p. 227 (1876) [Papers, t. iv. p. 231]. 

t "Tides and Waves," ^rt. 111. 

t "Tidal Researches," U.S. Coast Survey Rep. 1874, p. 154. 

§ "Rechorohes sur quelqnes points da syst^me dn monde," Mim. de VAcad. roy» des Sciences, 
1776 [1779] [Oeuvres, t. ix. ppw 187. . .]. 

II Oeuvres, t. ix. p. 218. The notation has been altered. 

^ It is remarkable that this argument is of a kind constantly employed by Airy himself in 
his researches on waves. 



334 



Tidal Waves 



[CHAP, vin 



The following table gives the numerical values of the coefficients of the 
several powers of v in the formula (19) for f/fl'", in the cases j3 = 40, 20, 10, 
5, 1, which correspond to depths of 7260, 14520, 29040, 58080, 290400 feet, 
respectively*. The last line gives the value of l^jH"' for v = 1, i.e, the ratio 
of the amplitude at the equator to its equilibrium-value. At the poles 
{y = 0), the tide has in all cases the equilibrium-value zero. 



• 


j8=40 


/3=20 


/3 = 10 , 

1 


/3=5 


^=1 ! 


y« 


+ 10000 


+ 1-0000 


+ 1-0000 


+ 1-0000 


+ 1-0000 


v^ 


+ 201862 


-0-2491 


+ 6-1915 


+ 0-7504 


+ 0-1062 


y« 


+ 101164 


- 1-4056 


+ 3-2447 


+01666 


+00039 


y8 


-131047 


-0-8594 


+ 0-7234 


+00157 


+00001 


vio 


- 15-4488 


-0-2541 


+ 00919 


+0-0009 


1 


y" 


- 7-4581 


-00462 


+0-0076 






v" 


- 2- 1975 


-00058 


+ 0-0004 






y" 


- 0-4501 


-0-0006 








y" . 


- 0-0687 










ySO 


- 0-0082 










yM 


- 0-0008 










y»* 


- 00001 












- 7-434 


-1-821 


+ 11-259 


+ 1-924 


+ 1110 

1 



We may use the above results to estimate the closeness of the approximation in each 
case. For example, when /3=40, Laplace finds 3^^= --000004ff'"; the addition to the 
disturbing force which is necessary to make the solution exact would then be - -00002^'"y^, 
and would therefore bear to the actual force the ratio - •00002y'®. 

It appears from (19) that near the poles, where v is small, the tides are 
in all cases direct. For sufficiently great depths, jS will be very small, and 
the formulae (17) and (19) then shew that the tide has everywhere sensibly 
the equilibrium- value, all the coefficients being small except the first, which 
is unity. As A is diminished, j8 increases, and the formula (17) shews that 
each of the ratios Nj will continually increase, except when it changes sign 
from + to — by passing through the value oo . No singularity in the 
solution attends this passage of Nj through oo , except in the case of iV^^ , 
since, as is easily seen, the product N^.^N^ remains finite, and the coefficients 
in (19) are therefore all finite. But when iV^j =. oo , the expression for { 
becomes infinite, shewing that the depth has then one of the critical values 
already referred to. 

The table above given indicates that for depths of 29040 feet, and 
upwards, the tides are everywhere direct, but that there is some critical 

* The first three cases were calculated by Laplace, Ix, ajUe p. 333 ; the last by Kelvin. The 
numbers relating to the third case have been slightly correcteil, in accordance with the computa- 
tions of Hough ; see p. 335. 



221-222] Hough's Theory 335 

depth between 29040 feet and 14520 feet, for which the tide at the equator 
changes from direct to inverted. The largeness of the second coefficient in 
the case j3 = 40 indicates that the depth could not be reduced much below 
7260 feet before reaching a second critical value. 

Whenever the equatorial tide is inverted, there must be one or more pairs 
of nodal circles ($ = 0), symmetrically situated on opposite sides of the 
equator. In the case of j3 = 40, the position of the nodal circles is pven by 
V = -95, or fi = 90° ± 18°, approximately*. 

222. The dynamical theory of the tides, in the case of an ocean covering 
the globe, with depth uniform along each parallel of latitude, has been greatly 
improved and developed by Hough f, who, taking up an abandoned attempt 
of Laplace, substituted expansions in spherical harmonics for the series of 
powers of fi (or v). This has the advantage of more rapid convergence, 
especially, as might be expected, in cases where the influence of the rotation 
is relatively small; and it also enables us to take account of the mutual 
attraction of the particles of water, which, as we have seen in the simpler 
problem of Art. 200, is by no means insignificant. 

If the surface-elevation ^, and the conventiontd equilibrium tide-height ^ 
(in which the effect of mutual attraction is not included), be expanded in 
series of spherical harmonics, thus 

^=SC„, ?=Sf„ (1) 

the complete expression for the disturbing potential will be 

2n H- 1 po 

cf. Art. 200. The series on the right-hand is to be substituted for ^ in the 
equations of Arts. 214. . . ; this will be allowed for if we write 

r = S (a„£„ - f„), (2) 

^^^'« "'" = i-2;^^„' (^) 

in modification of the notation of Art. 215 (5) or Art. 218.(4). 

In the oscillations of the * First Species,' the differential equation may be 
written 

If we assume _ 

f = SC„P„ (jtt). f=Sy«^„(/*). (5) 

wehave r = S (a„C„ - y„) P„ (jtt) (6) 

* For a fuller discnssion of these points reference may be made to the original investigation 
of Laplace, and to Kelyin's papers. 

t "On the Application of Harmonic Analysis to the Dynamical Theory of the Tides," Phil. 
Trans. A, t. clxzziz. p. 201, and t. czci. p. 139 (1897). See also Darwin's Papers, t. ii. p. 190. 



" = -^^(f« + 2lMrip^^«) = 



336 Tidal Waves [chap, vm 

Substituting in (4), and integrating between the limits — 1 and /i, we find 

(IP r/* 

S (o„C„ - y„) (1 - /**) ^ + Si8C,{(/« - 1) + (1 - /*«)}] ^ P»dM = 0. 

(7) 

Now, by known formulae of zonal harmonics''', 

and /^/»<^J^ = 2„ + 1 (^»« - ^-i) 

" 2» + 1 (2n + 3 \ d/tt d/4 / 2n-\\dii dfi 

1 <?Pn.H« 2 dP, 

(2» + 1) (2n + 3) d/[i (2n - 1) (2n + 3) d/j, 

^ (2n - 1) (2n + 1) d/x ' ' '^ ' 

dP 
Substituting in (7), and equating to zero the coefficient of (1 — jj.*) -^ 

we find 

Cn+t — LnCn + TKZ owoIT 1 \ ^«-» = "?>•• (10) 



(2n + 3) (2n + 5) "■"» " " ' (2n - 3) (2n - 1) "-» j9 

where ^" = n (n + 1) ■*■ (2n - .1) (2n + 3) ~ / ^^^^ 

The relation (10) will hold from n = 1 onwards, provided we put 

C_i = 0, Co = 0. 

The further theory is based substantially on the argument of Laplace, 
given in Art. 221; and the work follows much the same lines as in 
Arts. 216, 217, 221. 

In the free oscillations we have y„ = 0, and the admissible values of / 
are determined by the transcendental equation 

1 1 _ 

J. 5. 7«. 9 9. 11*. 13 - 

^« - -T:^ TT^&T = «' (12) 

1 1 



. 3. 5«. 7 7.9*. 11 - 

according as the mode is symmetrical or asymmetrical with respect to the 
equator. Alternative forms of the period equations are given by Hough, 

♦ See Todhunter, Functions of Laplace, dfc. c. v. ; Whittaker, Modem Analyais, Art. 117. 



» , 



222-223] 



Tide9 of Long Period 



337 



suitable for computation of the higher roots, and it is shewn that close 
approximations are given by the equations Zf^ » or 
'« (^ S p^ gh 2 



= l + n(n+l)-^ 1 



,.•(14) 



4a>* * ' " '" ' *' (V* 2n + 1 />o ^co^a^ (2n - 1) (2n + 3) 
except for the first two or three values of n. 

The following table gives the periods (in sidereal time) of the slowest 
symmetrical oscillation (i.e. the one in which the surface-elevation would 
vary as Pg (/n) if there were no rotation), corresponding to various depths*. 





Depth 


a* 


Period 


Period 


p 


(feet) 


4w* 




when w=0 


40 ' 






h. m. 


h. m. 


7260 


•44155 


18 3-5 


32 49 


20 


14520 


•62473 


15 110 


23 12 


10 


29040 


•92506 


12 28-6 


16 25 


6 


58080 


1-4785 


9 52- 1 


11 35 



The results obtained for the forced oscillations of the * First Species' are 
very similar to those of Art. 217. The limiting form of the long-period tides 
when a = shews the following results : 



1 

a 


pIp9= 


•181 


pIpo=o 


~ 1 

1 


Pole 


Equator 


Pole Equator 


40 


•140 


•426 


•154 -455 


20 


•266 


•551 






10 


•443 


•681 


•470 


•708 


5 


•628 


•796 


•651 

1 


•817 

1 



The second and third columns give the ratio of the polar and equatorial 
tides to the respective equilibrium- values f. The numbers in the fourth and 
fifth columns are repeated from Art. 217. The comparison shews the effect 
of the mutual gravitation of the water in reducing the amplitude. 

223. In the more general case, where symmetry about the axis is not 
imposed, the surface-elevation { is expanded by Hough in a series of tesseral 
harmonics of the type 

Pn (m) e^<'*+'*-^> (1) 

* The slowest asymmetrical mode has a much longer period. It involves a displacement of 
the centre of mass of the water, so that a correction would be necessary if the nucleus were free; 
cf. Art. 199. 

t The numbers are deduced from Hough's results. The paper referred to includes a discus- 
sion of many other interesting points, including an examination of cases of varying depth, with 
numerical illustrations. 



L. H. 



22 



338 



Tidal WafV€» 



[oHAP. vm 



In relation to tidal theory the most important cases are where the disturbing 
potential is of the form (1), with n = 2 and « = 1 or « = 2. 

The calculations are necessarily somewhat intricate*^, and it must suffice 
here to mention a few of the more interesting results, which will indicate 
how the gaps in the previous investigations have been filled. 

To understand the nature of the free oscillations, it is best to begin with 
the case of no rotation (co = 0). As o) is increased, the pairs of numerically 
equal, but oppositely-signed, values of a which were obtained in- Art. 199 
begin to diverge in absolute value, that being the greater which has the 
same sign with co. The character of the fundamental modes is also gradually 
altered. These oscillations are distinguished as ' of the First Class.' 

At the same time certain steady motions which are possible, without 
change of level, when there is no rotation, are converted into long-period 
oscillations with change of level, the speeds being initially comparable with 
CO. The corresponding modes are designated as 'of the Second Class 'f; 
cf. Art. 206. 

The following table gives the speeds of those modes of the First Class 
which are of most importance in relation to the diurnal and semi-diurnal 
tides, respectively, and the corresponding periods, in sidereal time. The last 
column repeats the corresponding periods in the case of no rotation, as 
calculated from the formula (15) of Art. 200. 





Second Species 

[* = 1] 


Third Species 
[*=2] 


Period 
when w=0 

h. m. 


Depth 
(feet) 




Period 
h. m. 




Period 
h. m. 


7260 
14620 
29040 
58080 


1-6337 
-0-9834 

1-8677 
-1-2460 

21641 
- 1-6170 

2-6288 
-21611 


14 41 
24 24 

12 61 
19 16 

11 6 
14 60 

9 8 
11 6 


1-3347 
-0-6221 

1-6133 
-0-8922 

1-9968 
-1-2855 

2-5536 
- 1-8575 


17 69 
38 34 

14 62 
26 54 

12 1 

18 40 

9 24 
12 55 


1 32 49 
1 23 12 
1 16 25 
1 11 36 



* A simplification is made by Love, "Notes on the Dynamical Theory of the Tides," Proc. 
Lond. Math, Soc. (2), t. zii. p. 309 (1913). He writes 

ad$ aBm0d<f>* a8mdd<f> add* 

of. Art. 154 (1). The values of x> ^ ai^ expanded in series of spherical harmonics. 

f These two classes of oscillations have been already encowitered in the plane problem of 
Art. 212. 



228] Diurnal and Semi'diumal Tides 339 

The quickest oscillation of the Second Glass has in each case a period of 
over a day ; and the periods of the remainder are very much longer. 

As regards the forced oscillations of the 'Second Species,' Laplace's 
conclusion that when a = a>, exactly, the diurnal tide vanishes in the case of 
uniform depth, still holds. The computation for the most important lunar 
diurnal tide, for which a/co = '92700, shews that with such depths as we have 
considered the tides are small compared with the equilibrium heights, and 
are in the main inverted. 

Of the forced oscillations of the 'Third Species,' we may note first the 
case of the solar semi-diurnal tide, for which a = 2ai with sufficient accuracy. 
For the four depths given in our tables, the ratio of the dynamical tide-height 
to the conventional equilibrium tide-height at the equator is found to be 

+ 7-9548, - 1-5016, - 23487, -f 2-1389, 
respectively. 

**The very large coefficients which appear when hg/4:a)^d^ = -^ indicate 
that for this depth there is a period of free oscillation of semi-diurnal type 
whose period differs but slightly from half-a-day. On reference to the 
tables ... it will be seen that we have, in fact, evaluated this period as 
12 hours 1 minute, while for the case hg/4o)^a^ == tV ^® have found a period 
of 12 hours 5 minutes*. We see then that though, when the period of 
forced oscillation differs from that of one of the types of free oscillation by as 
little as one minute, the forced tide may be nearly 250 times as great as the 
corresponding equilibrium tide, a difference of 5 minutes between these 
periods will be sufficient to reduce the tide to less than ten times the 
corresponding equilibrium tide. It seems then that the tides will not tend 
to become excessively large unless there is very close agreement with the 
period of one of the free oscillations. 

"The critical depths for which the forced tides here treated of become 
infinite are those for which a period of free oscillation coincides exactly with 
12 hours. They may be ascertained by putting [a = 2co] in the period- 
equation for the free oscillations and treating this equation as an equation 
for the determination of A The two largest roots are. . . , and the corre- 
sponding critical depths are about 29,182 feet and 7375 feet 

"It will be seen that in three cases out of the four here considered the effect 
of the mutual gravitation of the waters is to increase the ratio of the tide to 
the equilibrium tide [cf. Art. 221]. In two of the cases the sign is also re- 
versed. This of course results from the fact that whereas when [p/pi = 0* 18093] 
one of the periods of free oscillation is rather greater than 12 hours, when 
[pIpi = 0] the corresponding period will be less than 12 hoursf." 

* [Belonging to a mode which comee next in sequence to the one haying a period of 17 h. 59 m.] 
t Hough, Phil. Trans, A, t. ozoi. pp. 178, 179. 

22—2 



340 Tidal Waves [chap, vni 

Hough has also computed the lunar semi-diurnal tides for which 

^ = 0-96350. 

For the four depths aforesaid the ratios of the equatorial tide-heights to their 
equilibrium- values are found to be 

- 2-4187, - 1-8000, + 11-0725, + 1-9225, 
respectively. 

''On comparison of these numbers with those obtained for the solar 
tides. . . , we see that for a depth of 7260 feet the solar tides will be direct 
while the lunar tides will be inverted, the opposite being the case when the 
depth is 29,040 feet. This is of course due to the fact that in each of these 
cases there is a period of free oscillation intermediate between twelve solar 
(or, more strictly, sidereal) hours and twelve lunar hours. The critical depths 
for which the limar tides become infinite are found to be 26,044 feet and 
6448 feet. 

** Consequently this phenomenon will occur if the depth of the ocean be 
between 29,182 feet and 26,044 feet, or between 7375 feet and 6448 feet. 
An important consequence would be that for depths lying between these 
limits the usual phenomena of spring and neap tides would be reversed, the 
higher tides occurring when the moon is in quadrature, and the lower at new 
and full moon*." 

The most recent contribution to the dynamical theory consists of two 
papers by Goldsbroughf, who has discussed the tides in an ocean of uniform 
depth limited by one or two parallels of latitude. In the case of a polar 
basin of angular radius 30^, for instance, he finds that for such depths as 
have been considered in Arts. 217, 221 the long-period tides and the semi- 
diurnal tides do not deviate very widely from the values given by the equi- 
librium theory, when this is corrected as explained in the Appendix. The 
case is however very different with the diurnal tides, which vary considerably 
with the size of the basin and the depth, and are as a rule considerable, whereas 
we have seen that in a uniform ocean covering the globe they are negligible. 

In the case of an equatorial belt, the long-period tides again approximate 
to the equilibrium- values, whilst the diurnal and semi-diurnal deviate widely, 
to an extent which varies considerably with the position of the boundaries. 

The more difficult case of an ocean bounded by two meridians has not yet 
been investigated, except on the supposition that the angular velocity of 

* Hough, lx,j where reference is made to Kelvin's Popular Lectures and Addresses, London, 
1894, t. ii. p. 22 (1868). 

t "The Dynamical Theory of the Tides in a Polar Basin,** Proc Lond. Math. 8oc. (2), t. ziv. 
p. 31 (1913); "The Dynamical Theory of the Tides in a Zonal Ocean,** ibid. p. 207 (1914). 



223-224] Lag of the Tides 341 

rotation is small compaied with the speed of the free oscillations*. The 
plane problem of a circular sector would appear to be somewhat simpler, but 
has not been solved. 

224. It is not easy to estimate, in any but the most general way, the 
extent to which the foregoing conclusions of the dynamical theory would 
have to be modified if account could be taken of the actual configuration of 
the ocean, with its irregular boundaries and irregular variation of depthf . 
One or two points may however be noticed. 

In the first place, the formulae (17) of Art. 206 would lead us to expect 
for any given tide a phase-difference, variable from place to place, between 
the tide-height and the disturbing force:|:. Thus, in the case of the lunar 
semi-diurnal tides, for example, high- water or low-water need not synchronize 
with the transit of the moon or anti-moon across the meridian. More 
precisely, in the case of a disturbing force of given type for which the 
equilibrium tide-height at a particular place would be 

f SB a cos cr^, (1) 

the dynamical tide-height will be 

J = ii cos (cr« - €), (2) 

where the ratio Aja^ and the phase-diSerence €, will be functions of the 
speed a. 

Again, consider the superposition of two oscillations of the same type but 
of slightly different speeds, e.g. the lunar and solar semi-diurnal tides. If 
the origin of t be taken at a syzygy, we have 

f = a cos (jt-\- a' cos a% (3) 

and C^ Acqs {at — e) -\- A' cos {at — e') (4) 

This may be written 

^= {A-\- A' cos^) cos {ai — e) -\- A' sin^ sin {ai — e), (5) 

where <f> =^ {a - (/)t - e -{- e (6) 

If the first term in the second member of (4) represents the lunar, and the 
second the solar tide, we shall have a < a, and A > A'. If we write 

A + A' co8(f> = C cos a, il' sin ^ = C sin a, (7) 

we get f = C cos (<rf — € — a), (8) 

where C = {A^ + 2 A A' cos <f> -f- A'^)^ , a = tan-i . f ' !f "^ i . • - (9) 

A. ^~ A. COS u) 

* Rsyleigh, PnK. Boy. Boc A, t. Ixzxii. p. 448 (1909) [Pa/pen, t. ▼. p. 497]. 

t As to the general mathematioal problem reference may be made to Poinoar6, **Snr I'^ni- 
libre et les mouyements dee men,*' LumviUe (5), t. ii. pp. 57, 217 (1896), and to his Lefona de 
m^canique UUaU, t. iiL 

% This U illustrated by the canal problem of Art. 184. 



342 Tidal Waves [chap, vra 

This may be described as a simple-harmonic oscillation of slowly varying 
amplitude and phase. The amplitude ranges between the limits A ± A\ 
whilst a may be supposed to lie always between ± \7t. The 'speed' must 
also be regarded as variable, viz. we find 

da __ o-^» 4- (o- + a) A A' cos<^ + aA'^ 
"" dt A*'h2AA'coB<f> + A'^ ^^^^ 

This ranges between 

Aa + AW , Aa-AV ,,,,^ 

A-TA'- ^^* -Z3^' (^^) 

The above is the well-known explanation of the phenomena of the spring- 
and neap- tides t; but we are now concerned further with the question of 
phase. On the equilibrium theory, the maxima of the amplitude C would 
occur whenever 

{a — a)t = 2mr, 

where n is integral. On the dynamical theory the corresponding times of 
maximum are given by 

(a - a) <-(€'-€) = 2n7r, 

i.e. the dynamical maxima follow the statical by an interval | 

(€' - e)/(a' - a). 

If the difference between a and a were infinitesimal, this would be equal to 
defda. 

The fact that the time of high-water, even at syzygy, may follow or 
precede the transit of the moon or anti-moon by an interval of several hours 
is well known §. The interval, when reckoned as a retardation, is, moreover, 
usually greater for the solar than for the lunar semi-diurnal tide, with the 
result that the spring-tides are in many places highest a day or two after 
the corresponding syzygy. The latter circumstance has been ascribed || to 
the operation of Tidal Friction (for which see Chapter xi.), but it is evident 
that the phase-difierences which are incidental to a complete dynamical 
theory, even in the absence of friction, cannot be ignored in this connection. 
There is reason to believe that they are, indeed, far more important than 
those due to the latter cause. 

Lastly, it was shewn 'in Arts. 206, 217 that the long-period tides may 
deviate very considerably from the values given by the equihbrium theory, 

* Helmholtz, Lfhre von den Tonempfindungen (2« Aufi.), Braunaohweig, 1870, p. 622. 

t Cf . Thomaon and Tait. Art. 60. 

} This inteiral may of oonrse be negative. 

§ The values of the retardations (which we have denoted by e) for the various tidal com- 
ponents, at a number of ports, are given by Baird and Bandn, "Results of the Harmonic 
Analysis of Tidal Observations," Proc, B. S, t. xxxix. p. 135 (1885), and Darwin, ♦^Second 
Series of Results. . . ," Proc. JR. S, t. xlv. p. 666 (1889). 

II Airy, "Tides and Waves," Art. 459 



224-225] Stability of the Ocean 343 

0¥H[ng to the possibility of certain steady motions in the absence of disturbance. 
It has been pointed out by Bayleigh* that these steady motions may be 
impossible in certain cases where the. ocean is limited by perpendicular 
barriers. Referring to Art. 214 (6), it appears that if the depth h be 
imiform, ^ must (in the steady motion) be a function of the co-latitude 6 
only, and therefore by (4) of the same Art., the eastward velocity v must be 
uniform along each parallel of latitude. This is inconsistent with the existence 
of a perpendicular barrier extending along a meridian. The objection would 
not necessarily apply to the case of a sea shelving gradually from the central 
parts to the edgef . 

225. We may complete the investigation of Art. 200 by a brief notice 
of the question of the stability of the ocean, in the case of rotation. 

It has been shewn in Art. 205 that the condition of secular stability is 
that V — Tq should be a minimum in the equilibrium configuration. If we 
neglect the mutual attraction of the elevated water, the application to the 
present problepi is very simple. The excess of the quantity F — Tq ^^er its 
undisturbed value is evidently 




-\a}^w^)dz'^dS, (1) 

where ^ denotes the potential of the earth's, attraction, 85 is an element of 
the oceanic surface, and the rest of the notation is as before. Since "9 — Joi^td* 
is constant over the undisturbed level {z = 0), its value at a small altitude z 
may be taken to he gz -\- const., where, as in Art. 213, 



=[ 



I (^ - W^^) 



(2) 



Since SS^dS = 0, on account of the constancy of volume, we find from (1) that 
the increment of F — To is 

iM'dS (3) 

This is essentially positive, and the equilibrium is therefore secularly stable :|:. 

It is to be noticed that this proof does not involve any restriction as to 
the depth of the fluid, or as to smallness of the ellipticity, or even as to 
symmetry of the undisturbed surface with respect to the axis of rotation. 

If we wish to take into account the mutual attraction of the water, the 
problem can only be solved without difficulty when the undisturbed surface 
is nearly spherical, and we neglect the variation of g. The question (as to 
secular stability) is then exactly the same as in the case of no rotation. 

♦ "Note on the Theory of the Fortnightly Tide," PhU, Mag, (6), t. v. p. 136 (1903) [Papers^ 
t. lY. p. 84]. 

t The theory of the limiting forms of long-period tides in oceans of various types is discussed 
by Proudman, Proc. Lond. Math. Soc. (2), t. xiii. p. 273 (1913). 

t Cf . Laplace. Mdcaniqt^e OiUate, Livre 4me, Arts. 13, 14. 



344 Tidal Waves [chap, vin 

The calculation for this case will find an appropriate place in the next 
chapter. The result, as we might anticipate from Art. 200, is that the 
necessary and sufficient condition o£ stability of the ocean is that its density 
should be less than the mean density of the earth*. 

226. This is perhaps the most suitable occasion for a few additional 
remarks on the general question of stability of dynamical systems. We 
have in the main followed the ordinary usage which pronounces a state of 
equilibrium, or of steady motion, to be stable or unstable according to the 
character of the solution of the approximate equations of disturbed motion. 
If the solution consists of series of terms of the type Ce***, where all the 
values of A are pure imaginary (i.e. of the form ia), the undisturbed state is 
usually reckoned as stable ; whilst if any of the A's are real, it is accounted 
unstable. In the case of disturbed equilibrium^ this leads algebraically to 
the usual criterion of a minimum value of F as a necessary and sufficient 
condition of stability. 

It has in recent times been questioned whether this conclusion is, from 
a practical point of view, altogether warranted. It is pointed out that since 
Lagrange's equations become less and less accurate as the deviation from the 
equilibrium configuration increases, it is a matter for examination how far 
rigorous conclusions as to the ultimate extent of the deviation can be drawn 
from themf. 

The argument of Dirichlet, which establishes that the occurrence of 
a minimum value of F is a sufficient condition of stability, in any practical 
sense, has already been referred to. No such simple proof is available to 
shew without qualification that this condition is necessary. If, however, we 
recognize the existence of dissipative forces, which are called into play by 
any motion whatever of the system, the conclusion can be drawn as in 
Art. 205. 

A little consideration will shew that a good deal of the obscurity which 
attaches to the question arises from the want of a sufficiently precise 
mathematical definition of what is meant by 'stability.' The difficulty 
is encountered in an aggravated form when we pass to the question of 
stability of motion. The various definitions which have been propoimded 
by difierent writers are examined critically by Klein and Sommerfeld in 
their book on the theory of the top J. Rejecting previous definitions, they 
base their criterion on the character of the changes produced in the jxUh of 
the system by small arbitrary disturbing impulses. If the undisturbed path 
be the limiting form of the disturbed path when the impulses are indefinitely 

♦ Cf. Laplace, M4canique Cdeste, Livre 4™®, Arts. 13, 14. 

f See papers by Liapounofif and Hadamard, Lumville (5), t. iii. (1897). 

} Ueber die Theorie des Kreisds. Leipzig, 1897 . . . , p. 342. 



\ 



225-226] Dynamical Stability 345 

diminished, it is said to be stable, but not otherwise. For instance, the 
vertical fall of a particle under gravity is reckoned as stable, although for 
a given impulsive disturbance, however small, the deviation of the particle's 
position at any time t from the position which it occupied in the original 
motion increases indefinitely with t. Even this criterion, as the writers 
referred to themselves recognize, is not free from ambiguity imless the phrase 
^limiting form,' as applied to a path, be strictly defined. It appears moreover 
that a definition which is analytically precise may not in all cases be easy to 
reconcile with geometrical prepossessions*. 

The foregoing considerations have reference, of course, to the question 
of 'ordinary' stability. The more important theory of 'secular' stability 
{Alt, 206) is not afiected. We shall meet with the criterion for this, under 
a somewhat modified form, at a later stage in our subject f* 

* Some good illustrations are famished by Particle Dpiffcmics. Thus a particle moving in a 
circle about a centre of force ▼ar3ang inversely as the cube of the distance will if slightly disturbed 
either fall into the centre, or recede to infinity, after describing in either case a spiral with an 
infinite number of convolutions. Each of these spirals has, analytically, the circle as its 
* limiting form,' although the motio>n in the latter is most naturally described as unstable. 
Of. Korteweg, Wiener Ber. May 20, 1886. 

A narrower definition has been given by Love, and applied by Bromwich to several dynamical 
and hydrodynamioal problems; see Proc. Land, Math. Soc t. zxxiii. p. 325 (1901). 

t This summary is taken substantially from the Art. "Dynamics, Analytical,*' in Encyc. 
Brit, 10th ed., t. uvii. p. 566 (1902), and 11th ed., t. viu. p. 756 (1910). 



4 
* 

I 
I 

I 



^ 



APPENDIX 

TO CHAPTER VIII 

ON TIDE-GENERATING FORCES 

a. If, in the annexed figure, and C be the centres of the earth and of the disturbing 
body (say the moon), the potential of the moon's attraction at a point P near the earth's 
surface will be -yJUfCP, %«kcre M denotes the moon's mass, and y the gravitation- 
constant. If we put OC=D, OP=r, «fid denote the moon's (geocentric) zenith-distance 
at P, viz. the angle POC, by 3, this potential is equal to 

yM 

(Z)«-2rDcos3 + r«)i* 




We require, however, not the absolute accelerative effect at P. buiVthe acceleration 
relative to the earth. Now the moon produces in the whole mass ol the earth an 
acceleration yM/D^* parallel to OC, and the potential of a uniform field o^ force of thip 
intensity is evidently 

y3f 



D« 



. r cos S, 



Subtracting this from the former result we get, for the potential of the relative V^^^^^^^^ 
atP, 

^ + ^ . r cos ^ 



o= - 



(1) 



{D^ -2rD COS S+r^)i ^* 
This function Q is identical with the * disturbing-function' of planetary theory. 

Expanding in powers of r/D, which is in our case a small quantity, and retainin]| only 
the most importeuit term, we find 



o=*^ 



i ^(i-cos«5). 



Considered as a function of the position of P, this is a zonal harmonic of the 8< 
degree, with OC as axis. 



5) 
md 



• The effect of this is to produce a monthly inequality in the motion of the earth's cenV'^ 
about the sun. The amplitude of the inequality in radius vector is about 3000 miles; that I of 
the inequality in longitude is about 7"; see Laplace, M4canique COesie, livre &^, Art. 30, apd 
Livre 13"»% Art. 10. 



APR] Equilibrium Theory 847 

The reader will easily verify that, to the order of approximation adopted, Q is equal to 
the joint potential of two masses, each equal to ^if, placed, one at C, and the other at a 
point C in CO produced such that OC =0C*. 

b. In the 'equilibrium-theory' of the tides it is assumed that the free surface takes 
at each instant the equilibrium-form which might be maintained if the disturbing body 
were to retain unchanged its actual position relative to the rotating earth. In other 
words, the free surface is assumed to be a level-surface under the combined action of 
gravity, of centrifugal force, and of the disturbing force. The equation to this level- 
surface is 

* -^•oj* +Q =const (3) 

where <i> is the angular velocity of the rotation, w denotes the distance of any point from 
the earth's axis, and ^ is the potential of the earth's attraction. If we use square 
brackets [ ] to distinguish the values of the enclosed quantities at the undisturbed level, 
and denote by f the elevation of the water above this level due to the disturbing 
potential 12, the above equation is equivalent to 



[* - Ja)«w«] +r^(* -io)"w«)|f +12 = const., (4) 



approximately, where d/dz is used to indicate a space-differentiation along the normal 
outwards. The first term is of course constant, and we therefore have 

C--^+C (6) 

where, as in Art. 213, ^=r^(* -i»*ar«)1 (6) 

Evidently, g denotes the value of 'apparent gravity'; it will of course vary more or less 
with the position of P on the earth's surface. 

It is usual, however, in the theory of the tides, to ignore the slight variations in the 
value of g, and the effect of the eUipticity of the undisturbed level on the surface- value 
of a. Putting, then, r=:a, g=yE/a\ where E denotes the earth's mass, and a the mean 
radius of the surface, we have, from (2) and (6), 

f=jy(cos»^-J)+C (7) 



where ^=|.|. («)•.., 



(8) 



as in Art. 180. Hence the equilibrium-form of the free surface is a harmonic spheroid of 
the second order, of the zonal type, whose axis passes through the disturbing body. 

C. Owing to the diurnal rotation, and also to the orbital motion of the disturbing 
body, the position of the tidal spheroid relative to the earth is continually changing, 
so that the level of the water at any particular place will continually rise and fall. 
To anal3rse the character of these changes, let be the co-latitude, and <^ the longitude, 
measured eastward from some fixed meridian, of any place P, and let Abe the north-polar- 
distance, and a the hour-angle west of the same meridian, of the disturbing body. 
We have, then, 

cos S =co8 A cos B +sin A sin cos (a +<^), (9) 

* Thomson and Tait, Art. 804. These two fictitious bodies are designated as 'moon' and 
'anti-moon,' respectively. 



348 On Tide-Grcnerating Forces [chap, vin 

and thenoe, by (7), 

C=f£r(ooe«A-i)(oo8«^-i) 

+ ^ JJ sin 2a sin 2^ COS (a + 0) 

+i£r8m« Asin* ^ ooe 2 (a +<^) +C (10) 

Each of these terms may be regarded as representing a partial tide, and the results 
superposed. 

Thus, the first term is a zonal harmonic of the second order, and gives a tidal spheroid 
symmetrical with respect to the earth's axis, having as nodal lines the parallels for which 
cos* B=i,otO= 90° ±, 35° 16^ The amount of the tidal elevation in any particular latitude 
varies as cos' A - J . In the case of the moon the chief fluctuation in this quantity has 
a period of about a fortnight; we have here the origin of the * lunar fortnightly' or 
*declinationar tide. When the sun is the disturbing body, we have a 'solar semi-annual' 
tide. It is to be noticed that the mean value of cos' A - ^ with respect to the time is not 
zero, so that the inclination of the orbit of the disturbing body to the equator involves as 
a consequence a permanent change of mean level. Of. Art. 183. 

The second term in (10) is a spherical harmonic of the type obtained by putting n =2, 
« = 1 in Art. 86 (7). The corresponding tidal spheroid has as nodal lines the meridian 
which is distant 90° from that of the disturbing body, and the equator. The disturbance 
of level is greatest in the meridian of the disturbing body, at distances of 45° N. and S. of 
the equator. The osciUation at any one place goes through its period with the hour- 
angle, a, i.e. in a lunar or solar day. The amplitude is, however, not constant, but varies 
slowly with A, changing sign when the disturbing body crosses the equator. This term 
accounts for the lunar and solar *diumar tides. 

The third term is a sectorial harmonic (n=2, «=2), and gives a tidal spheroid having 
as nodal lines the meridians which are distant 46° E. and W. from that of the disturbing 
body. The oscillation at any place goes through its period with 2a, t.e. in half a (lunar or 
solar) day, and the amplitude varies as sin' A, being greatest when the disturbing body is 
on the equator. We have here the origin of the lunar and solar * semi-diurnal' tides. 

The 'constant' O is to be determined by the consideration that, on account of the 
invariability of volume, we must have 

/Jf(Mf=0 (11) 

where the integration extends over the surface of the ocean. If the ocean cover the 
whole earth we have C =0, by the general property of spherical surface-harmonics quoted 
in Art. 87. It appears from (7) that the greatest elevation above the undisturbed level is 
then at the points .9=0, ^ = 180°, t.e. at the points where the disturbing body is in 
the zenith or nadir, and the amount of this elevation is f if. The greatest depression is at 
places where ^ =90°, i.e. the disturbing body is on the horizon, and is f JST. The greatest 
possible range \a therefore equal to H, 

In the case of a limited ocean, C does not vanish, but has at each instant a definite 
value depending on the position of the disturbing body relative to the earth. This value 
may be easily written down from equations (10) and (11); it is a sum of spherical 
harmonic functions of A, a, of the second order, with constant coefficients in the form of 
surface-integrals whose values depend on the distribution of land and water over the 
globe. The changes in the value of C, due to relative motion of the disturbing body, 
give a general rise and fall of the free surface, with (in the case of the moon) fortnightly, 
diurnal, and semi-diurnal periods. This 'correction to the equilibrium-theory' as usuaUy 



APP.] Harmonic Analysis 349 

presented, was first fully investigated by Thomson and Tait*. The necessity for a 
oorreotion of the kind, in the case of a limited sea, had however been recognized by 
D. BemouUif. 

The correction has an influence on the time of high water, which is no longer synchronous 
with the maximum of the disturbing potential. The interval, moreover, by which high 
water is accelerated or retarded differs from place to place:(. 

d. We have up to this point neglected the mutual attraction of the particles of the 
water. To take this into account, we must add to the disturbing potential Q the 
gravitation-potential of the elevated water. In the case of an ocean covering. the earth, 
the correction can be easily applied, as in Art. 200. If we put n=2 in the formulae of 
that Art., the addition to the value of Q is -^p/po'gCi ^^^ ^® thence find without 
difficulty 

^=n«;<°"*'-*^ <''> 

It appears that all the tides are increasedf in the ratio (1 -fp/pu)~^* ^ ^^ assume 
p/po = *18, this ratio is 1-12. 

e. So much for the equilibrium-theory. For the purposes of the kinetic theory 
of Arts. 21^224, it is necessary to suppose the value (10) of ^ to be expanded in a 
series of simple-harmonic functions of the time. The actual expansion, taking account of 
the variations of A and a, and of the distance 2> of the disturbing body (which enters 
into the value of H), is a somewhat complicated problem of Physical Astronomy, into 
which we do not enterf . 

Disregarding the constant C, which disappears in the dynamical equations (1) of 
Art. 215, the constancy of volume being now secured by the equation of continuity (2), it 
is easily seen that the terms in question will be of three distinct types. 

First, we have the tides of long period, for which 

C=H' (coe« e -J) . cos {(Tt +€) (13) 

The most importeutit tides of this class are the * lunar fortnightly' for which, in degrees 
per mean solar hour, o- = 1°'098, and the *solar-annuar for which 0-= 0^*082. 

Secondly, we have the diurnal tides, for which 

C=H" BiaBoo&e.co6{(rt-\-<l) +«), (14) 

where <r differs but little from the angular velocity o> of the earth's rotation. These 
include the *lunar diurnal' (<r = 13**-W3), the 'solar diurnal' (a = 14°-969), and the Muni- 
solar diurnal' (cr =» = 15^-041). 

* Natural Philosophy, Art. 808; see also Darwin, "On the Correction to the Equilibrium 
Theory of the Tides for the Continents," Proc. Roy. 80c. April 1, 1886 [Papers, t. i. p. 328]. It 
appears as the result of a numerical calculation by Prof. H. H. Turner, appended to this paper, 
that with the actual distribution of land and water the correction is of little importance. 

t Traits aur U Flux ei Reflux de la Mer, c. xL (1740). This essay, as well as the one by 
Maclaurin cited on p. 300, and another on the same subject by Euler, is reprinted in Le Seur and 
Jacquier's edition of Newton's Principia, 

X Thomson and Tait, Art. 810. The point is illustrated by the formula (3) of Art. 184 supra. 

§ Reference may be made to lAplaoe, Micaniqite Celeste, Livre 13"**, Art. 2, and to Darwin*s 
Papers, t. i. 



850 On Tide-Generating Forces [chap, vm 

Lastly, we have the semi-diurnal tides, for which 

f =fr'" sin* e . oosM +2<^ +*), . . (16)* 

where <r differs but little from 2<o. These include the * lunar semi-diurnal' (<r =28^-984), 
the * solar semi-diurnal* (<r =30°), afid the *luni-solar semi-diurnal* (tr =2<o =30^-082). 

For a complete enumeration of the more important partial tides, and for the values of 
the coefficients H\ H"y H"' in the several cases, we must refer to the investigations of 
Darwin, already cited. In the Harmonic Analysis of Tidal Observations, which is the 
special object of these investigations, the only result of dynamical theory which is made 
use of is the general principle that the tidal elevation at any place must be equal to the 
sum of a series of simple-harmonic functions of the time, whose periods are the same as 
those of the several terms in the development of the disturbing potential, and are therefore 
known d 'priori. The amplitudes and phases of the various partial tides, for any particular 
port, are then determined by comparison with tidal observations extending over a 
sufficiently long period f. We thus obtain a practically complete expression which can be 
used for the systematic prediction of the tides at the port in question. 

f. One point o( special interest in the Harmonic Analysis is the determination of the 
long-period tides. It has been already stated that under the influence of dissipative 
forces these must tend to approximate more or less closely to their equilibrium values. 
Unfortunately, the only long-period tide, whose coefficient can be inferred with any 
certainty from the observations, is the lunar fortnightly, and it is at least doubtful whether 
the dissipative forces are sufficient to produce in this case any great effect in the direction 
indicated. Hence the observed fact that the fortnightly tide has less than its equilibrium 
value does not entitle us to make any inference as to elastic yielding of the solid body of 
the earth to the tidal distorting forces exerted by the moon %. 

* It is evident that over a small area, near the poles, which may be treated as sensibly plane, 
the formulae (14) and (15) make 

^« r COB (<r< + 0-1-6), and f a r * cos (<ri -i- 2^ -|- e), 

respectively, where r, w are plane polar co-ordinates. These forms have been used by anticipation 
in Arts. 211, 212. 

f It is of interest to note, in connection with Art. 187, that the tide-gauges, being situated 
in relatively shallow water, are sensibly affected by certain tides of the second order, which there- 
fore have to be taken account of in the general scheme of Harmonic Analysis. 

} Darwin, Uc, ante p. 323. See, however, the paper by Bayleigh cited on p. 343 ante. 



CHAPTER IX 

SURFACE WAVES 

227. We have now to investigate, as far as possible, the laws of wave- 
motion in liquids when the vertical acceleration is no longer neglected. The 
most important case not covered by the preceding theory is that of waves 
on relatively deep water, where, as will be seen, the agitation rapidly 
diminishes in amplitude as we pass downwards from the surface; but it 
will be imderstood that there is a continuous transition to the state of 
things investigated in the preceding chapter, where the horizontal motion 
of the fluid was sensibly the same from top to bottom. 

We begin with the oscillations of a horizontal sheet of water, and we will 
confine ourselves in the first instance to cases where the motion is in two 
dimensions, of which one {x) is horizontal, and the other (y) vertical. The 
elevations and depressions of the free surface wiU then present the appearance 
of a series of parallel straight ridges and furrows, perpendicular to the 
plane xy. 

The motion, being assumed to have been generated originally from rest 
by the action of ordinary forces, will necessarily be irrotational, and the 
velocity-potential <f> will satisfy the equation 

al + a? = *' (1) 

with the condition ^ "^^ (2) 

at a fixed boundary. 

To find the condition which must be satisfied at the free surface 
{p r= const.), let the origin be taken at the undisturbed level, and let Oy 
be drawn vertically upwards. The motion bdng assumed to be infinitely 
small, we find, putting Cl=^gy 'm the formula (4) of Art. 20, and neglecting 
the square of the velocity (g), 



M-»»+^('). 



(3) 



352 Surfa^^e Waves [chap, ix 

Hence if t\ denote the elevation of the surface at time i above the point {Xy 0)» 
we shall have, since the pressure there is uniform, 

-^[ti... '« 

provided the function F (^), and the additive constant, be supposed merged 
in the value of 3<^/3^ Subject to an error of the order already neglected, 
this may be written 

-HIL '^' 

Since the normal to the free surface makes an infinitely small angle 
(drj/dx) with the vertical, the condition that the normal component of the 
fluid velocity at the free surface must be equal to the normal velocity of the 
surface itself gives, with sufficient approximation, 

l-[IL '^' 

This is in fact what the general surface condition (Art. 9 (3)) becomes, if 
we put F {x, y^ z, t) = y — t), and neglect small quantities of the second order. 

Eliminating rj between (5) and (6), we obtain the condition 

dt* ^dy ' ^ ' 

to be satisfied when y = 0. This is equivalent to Dp/Dt = 0. 

In the case of simple-harmonic motion, the time-factor being e*<'*+*^, this 



condition becomes 



<^<f> = 9^ (8) 



228. Let us apply this to the free oscillations of a sheet of water, or 
a straight canal, of uniform depth h, and let us suppose for the present that 
there are no limits to the fluid in the direction of x, the fixed boundaries, if 
any, being vertical planes parallel to xy. 

Since the conditions are uniform in respect to x, the simplest supposition 
we can make is that ^ is a simple-harmonic function of x ; the most general 
case consistent with the above assumptions can be derived from this by 
superposition, in virtue of Fourier's Theorem. 

We assume then 

<f> = P cos Jcx, e*<'*+'>, , (1) 

where P is a function of y only. The equation (1) of Art. 227 gives 

^-**^ = 0. (2) 

whence P = Ae^^ + Be-^^ (3) 



227-228] Standing Waves 353 

The condition of no vertical motion at the bottom is 3^/3y = for y = — A, 

whence 

ile-** = Be**, = JC, say. 

This leads to <f> = C cosh k{y -^ h) cos kx . e*<*'*+*^ (4) 

The valuie of a is then determined by Art. 227 (8), which gives 

or* = gk tanh kh (5) 

Substituting from (4) in Art. 227 (5), we find 

iaC 
T] = — cosh kh cos kx . e*^'*+'^ (6) 

9 
or, writing a = . cosh M, 

and retaining only the real part of the expression, 

7) = a cos fee . sin (o^ + €) (7) 

This represents a system of 'standing waves,' of wave-length A = 27r/i, 
and vertical amplitude a. The relation between the period (27r/or) and the 
wave-length is given by (5). Some numerical examples of this dependence 
are given on p. 357. 

In terms of a we have 

, qa cosh Jk (v + A) , / . v 
^ = -a coshM ^ cos to ■ coa (a< + e). (8) 

and it is easily seen from Art. 62 that the corresponding value of the stream- 
function is 

, flra sinh i (y + A) . , /^ . . //^v 

'f'a coshM ' ^*^-°"°('^ + ^) (^> 

If X, y be the co-ordinates of a particle relative to its mean position 
(x, y), we have 

it" dx' dJt'' dy' ^^' 

if we neglect the differences between the component velocities at the points 
(x, y) and (x -\-x,y + y), as being small quantities of the second order. Sub- 
• stituting from (8), and integrating with respect to t, we find 

cosh k(y + h) , J • /^ , \ 

X = — a . ^j, — sm kx . sm (ctt + e), 

smhM . .--. 

sinh k(y + h) , . , ^ . f 

y = a . , TT — ' cos kx . sm ((ji -f- c), 

smhM ^ 'V 

where a slight reduction has been effected by means of (5). The motion of 
each particle is rectilinear, and simple-harmonic, the direction of motion 
varjring from vertical, beneath the crests and hollows {kx = mrr), to horizontal, 
beneath the nodes (kx = {m + I) n). As we pas6 downwards from the surface 

L. H. 23 



354 



Surface Waves 



[chap. IX 



to the bottom the amplitude of the vertical motion diminishes from a cos he 
to 0, whilst that of the horizontal motion diminishes in the ratio cosh kh:\. 

When the wave-length is very small compared with the depth, kh is large, 
and therefore tanh kh= 1*. The formulae (11) then reduce to 

X = — ae^y sin kx . sin (at + c), y = ae^^ cos kx , sin {<ft + e), . . (12) 

with a^ = gk, .^ (13) 

The motion now diminishes rapidly from the surface downwards; thus at 
a depth of a wave-length the diminution of amplitude is in the ratio c"*' or 
1/535. The forms of the lines of (oscillatory) motion (iff = const.), for this 
case, are shewn in the annexed figure. 




In the above investigation the fluid is supposed to extend to infinity in 
the direction of x, and there is consequently no restriction to the value of k. 
The formulae also give, however, the longitudinal oscillations in a canal of 
finite length, provided k have the proper values. If the fluid be bounded by 
the vertical planes x = 0, x^l (say), the condition 3^/3x = is satisfied at 

both ends provided sin AZ = 0, or kl = mw; where m = 1, 2, 3, The 

wave-lengths of the normal modes are therefore given by the formula A = 2l/m. 
Cf. Art. 178. 

229. The investigation of the preceding Art. relates to the case of 
^standing' waves; it naturally claimed the first place, as a straightforward 
application of the usual method of treating the free oscillations of a system 
about a state of equilibrium. 

In the case, however, of a sheet of water, or a canal, of uniform depth, 
extending horizontally to infinity in both directions, we can, by super- 
position of two systems of standing waves of the same wave-length, obtain 
a system of progressive waves which advance unchanged with constant 
velocity. For this, it is necessary that the crests and troughs of one 
component system should coincide (horizontally) with the nodes of the other, 
that the amplitudes of the two systems should be equal, and that their 
phases should differ by a quarter-period. 

* This case may of course be more easily investigated independently. 



228^-229] Progressive Waves 355 

Thus if we put V^Vi^Vi* (1) 

where 7ji=^ a^kx cos <ft, 172 '= ^ ^^^ kx mi at^ (2) 

we get 7) = a sin {kx ± at), * (3) 

which represents an infinite train of waves travelling in the negative or 
positive direction of a;, respectively, with the velocity c given by 



c 



= ?=(|tanhM)*, (4) 



where the value of a has been substituted from Art. 228 (5). In terms of 
the wave-length (A) we have 



c 



-(£-*x)* <») 



When the wave-length is anything less than double the depth, we have 
tanh M = 1, sensibly, and therefore* 



c 



-©*-©* («) 



On the other hand when A is moderately large compared with h we have 
tanh kh » M, nearly, so that the velocity is independent of the wave-length, 
being given by 

c = {9h)\ 0) 

as in Art. 170. This result is here obtained on the assumption that the 
wave-profile is a curve of sines, but Fourier's Theorem shews that the 
restriction is now to a great extent unnecessary. 

It appears, on tracing the curve y « (tanh x)/x, or from the numerical 
table to be given presently, that for a given depth h the wave- velocity 
increases constantly with the wave-length, from zero to the asymptotic 
value (7). 

Let us now fix our attention, for definiteness, on a train of simple-harmonic 
waves travelling in the positive direction, i.e. we take the lower sign in (1) 
and (3). It appears, on comparison with Art. 228 (7), that the value of ^1 is 
deduced by putting € = Jtt, and subtracting Jtt from the value of fccf, and 
that of 7j2 by putting c = 0, simply. This proves a statement made above as 
to the relation between the component systems of standing waves, and also 
enables us to write down at once the proper modifications of the remaining 
formulae of the preceding Art. 

* Green, "Note on the Motion of Waves in Canals," Camb. Trans, t. vii. (1839) [Papers, 
p. 279]. 

t This is merely equivalent to a change of the origin from which x is measured. 

2a— 2 



356 , Surface Waves [chap, ix 



TIuis, we find, for the component displacements of a particle, 

cosh A (y + A) ,, ^, ^ 



sinh A (y + A) . ,, ^. 



(8) 



This shews that the motion of each particle is elliptic-harmonic, the 
period (27r/(7, = X/c) being that in which the disturbance travels over a wave- 
length. The semi-axes, horizontal and vertical, of the elliptic orbits are 

^ cosh kjy-jrh) ^^^ ^ sinh k{y-}-h) 
sinhM sinhA;A ' 

respectively. These both diminish from the surface to the bottom (y = — h), 
where the latter vanishes. The distance between the foci is the same for all 
the ellipses, being equal to acosechM. It easily appears, on comparison 
of (8) with (3), that a surface-particle is moving in the direction of wave- 
propagation when it is at a crest, and in the opposite direction when it is in 
a trough*. 

When the depth exceeds half a wave-length, c~** is very small, and the 
formulae (8) reduce to 

X = oe**' cos {Jcx — at), y = ae^^ sin {kx — at), (9) 

so that each particle describes, a circle, with constant angular velocity 

a, = (2Trgr/A)'t- The radii of these circles are given by the formula ae^^, 
and therefore diminish rapidly downwards. 

In the first table on the next page, the second colunm gives the values 
of sech kh corresponding to various values of the ratio A/A. This quantity 
measures the ratio of the horizontal motion at the bottom to that at the 
surface. The third column gives the ratio of the vertical to the horizontal 
diameter of the elliptic orbit of a surface-particle. The fourth and fifth 
columns give the ratios of the wave-velocity to that of waves of the same 
length on water of infinite depth, and to that of *long' waves on water of 
the actual depth, respectively. 

The tables of absolute values of periods and wave- velocities, which are also 
given on p. 357, are abridged from Airy's treatise J. The value of g adopted 
by him is 32* 16 ft./sec.«. 

The possibility of progressive waves advancing with imchanged form is 
limited, theoretically, to the case of uniform depth; but the numerical 

♦ The results of Arts. 228, 229, for the case of finite depth, were given, substantially, by Airy, 
"Tides and Waves," Arts. 160. . . (1846). 
t Green, l.c. ante p. 366. 
t "Tides and Waves," Arts. 169, 170. 



229] 



Numerical RemUs 



357 



results shew that a variation in the depth will have no appreciable influence, 
provided the depth everywhere exceeds (say) half the wave-length. 





1 
aechibA 


tanhibA 


cligk-^)^ 


c/to«* 


000 


1000 


0000 


0000 


1000 


•01 


•998 


•063 


•260 


•999 


•02 


•992 


•126 


•364 


•997 


•03 


•983 


•186 


•432 


•994 


•04 


•969 


•246 


•496 


•990 


•06 


•963 


•304 


•662 


•984 


•06 


•933 


•360 


•600 


•977 


•07 


•911 


•413 


•643 


•970 


•08 


•886 


•464 


•681 


•961 


•09 


•869 


•612 


•716 


•961 


•10 


•831 


•667 


•746 


•941 


•20 . 


•627 


•860 


•922 


•823 


•30 


•297 


•966 


•977 


•712 


•40 


•161 


•987 


•993 


•627 


•60 


•086 


•996 


•998 


•663 


•60 


•046 


•999 


•999 


•616 


•70 


•026 


1000 . 


1000 


•477 


•80 


•013 


1000 


1000 


•446 


•90 


•007 


1000 


1-000 


•421 


100 


•004 


1000 


1000 


•399 


00 


•000 


1000 


1000 


•000 



Depth of 






Length of wave, in feet 


water, 
in feet 


1 


10 


100 


1000 


10,000 










Period of WA-VA. in RARonclfl 


1 


0442 


1873 


17-646 


176-33 


1763-3 




10 


0442 


1-398 


6-923 


6680 


667-62 




100 


0-442 


1398 


4-420 


1873 


176-46 




1000 


0442 


1398 


4-420 


1398 


6923 




10,000 


0442 


1398 


4-420 


1398 


44-20 





1 

Depth of 


• 


Length of 


wave, in feet 

» 1 




water, 
in feet 


1 


10 


100 


1000 


10,000 


00 






Wave- velocity, 
L 6-339 i 6667 


in feet per sARond 




1 


2-262 


6-671 


6-671 


6-671 


10 


2-262 


7-154 


16-88 


17-92 


17-93 


17-93 


100 


2-262 


7164 


22-62 


63-39 


66-67 


66-71 


1000 


2-262 


7-164 


22-62 


71-64 


168-8 


179-3 


10,000 


2-262 


7-164 


22-62 


7164 


226-2 


6671 



358 Surface Waves [chap, ix 

We remark, finally, that the theoiy of progressiTe waves may be obtained, 
without the intermediary of standing waves, by assnming at once, in place 
of Art. 228 (1), 

if> = Pc'<'*-»*>. .'. (10) 

« 
The conditions to be satisfied by P are exactly the same as before, and we 

easily find, in real form, 

ly = a sin (fee — at), (11) 

. aacoshA;(y + A) ,, ^. ,^^. 

^°f coshit ^ C08(fa:-at). (12) 

with the same determination of a as before. From (12) all the preceding 
results as to the motion of the rudividual particles can be inferred without 
difficulty. 

230. The energy of a system of standing waves of the simple-harmonic 
type is easily found. If we imagine two vertical planes to be drawn at unit 
distance apart, parallel to a^, the potential energy per wave-length of the 
fluid between these planes is 



J 





Substituting the value of i) from Art. 228 (7), we obtain 

IgfM'^X . sin« (o< + e) (1) 

The kinetic energy is, by the formula (1) of Art. 61, 



*'/: [*i]...*'- 



Substituting from Art. 228 (8), and remembering the relation between q and 
i, we obtain 

Igpa^X . cos* (oi + c) (2) 

The total energy, being the sum of (1) and (2), is constant, and equal to 
\gpa^\. We may express this by saying that the total energy per unit area 
of the water-surface is \gpa^. 

A similar calculation may be made for the case of progressive waves, or 
we may apply the more general method explained in Art. 174. In either 
way we find that the energy at any instant is half potential and half kinetic, 
and that the total amount, per unit area, is \gp(^- In other words, the 
energy of a progressive wave-system of amplitude a is equal to the work 
which would be required to raise a stratum of the fluid, of thickness a, 
through a height \a. 



229-231] 



Energy 



369 



231. The theory of progressiye waves may be investigated, in a very 
compact manner, by the method of Art. 175*. 

Thus if <f>, if/ be the velocity- and stream-functions when the problem has 
been reduced to one of steady motion, we assume 

<f> + uff 



= - (a? + iy) + iae<*<*+<»J -i- iJ3e-<*<»+<>'>, 



whence 



c 



X - (ae-"*» - i8c*«') sin kx, 
r = ^ y 4- {ae-^v + pe^v) cos hx. 



(1) 



This represents a motion which is periodic in respect to x, superposed on 
a uniform current of velocity c. We shall suppose that ka and kp are small 
quantities; in other words, that the amplitude of the disturbance is small 
compared with the wave-length. 

The profile of the free surface must be a stream-line ; we will take it to 
be the line = 0. Its form is then given by (1), viz. to a first approximation 
we have 

y = (a H- j3) cos fee, (2) 

shewing that the origin is at the mean level of the surface. Again, at the 
bottom (y = — A) we must also have = const. ; this requires 

06** -i- j3c-** = 0. 
The equations (1) may therefore be put in the forms 



" = — a; -I- C cosh k{y -\- h)m.nkx, 

- = — y + C' sinh k{y -\- h) cos kx. 
c 

The formula for the pressure is 

— -^-»©*-(i)] 



(3) 



P 

P 



A 



= const. — fly — -o {1 — 2A;C cosh k{y + h) cos kx}, 

if we neglect k^CK Since the equation to the stream-line = is 

y = C sinh kh cos kx, 

approximately, we have, along this line, 

- = const. 4- {kc^ coth kh — g) y. 
P 



(*) 



* Rayleigh, l.c, ante p. 252. 



360 Surface Waves [chap, ix 

The condition for a free surface is therefore satisfied, provided 

4 , tanhM ,^. 

"=^*— HfcT (^) 

This determines the wave-length (27r/A;) of possible stationary undulations on 
a stream of given uniform depth A, and velocity c. It is easily seen that the 
value of kh is real or imaginary according as c is less or greater than {ghy. 

If we impress on everything the velocity — c parallel to x, we get 
progressive waves on still water, and (5) is then the formula for the wave- 
velocity, as in Art. 229. 

When the ratio of the depth to the wave-length is sufficiently great, the 
formulae (1) become 

^ = - « + j3e*y sin fee, ?^ = - y + jSe*" cos fee, (6) 

leading to 2 = const, -gy-%{l- 2kpe^y cos kx + A;«jS*e"«'} (7) 

If we neglect i*jS*, the latter equation may be written 

2 = const. + (Ac* - j) y + hop (8) 

Henceif ^'==1' (^) 

the pressure will be uniform not only at the upper surface, but along every 
stream-line »= const.* This point is of some importance ; for it shews that 
the solution expressed by (6) and (9) can be extended to the case of any 
number of liquids of different densities, arranged one over the other in 
horizontal strata, provided the uppermost surface be free, and the total depth 
infinite. And, since there is no limitation to the thinness of the strata, we 
may even include the case of a heterogeneous liquid whose density varies 
continuously with the depth. Cf. Art. 235. 

232. The method of the preceding Art. can be readily adapted to a 
number of other problems. 

1*. For example, to find the velocity of propagation of waves over the 
common horizontal boundary of two masses of fluid which are otherwise 
unlimited, we may assume 

5^ = - y + jSe*>' cos fee, ^ = - y + jSe-*>' cos fee, (1) 

where the accent relates to the upper fluid. For these satisfy the condition 
of irrotational motion, V*^ = ; and they give a uniform velocity c at a great 

* This conclusion, it must be noted, is limited to the case of infinite depth. It was first 
remarked by Poisson, Lc. post p. 373. 



231-232] Oscillations of Superposed Liquids 361 

distance above and below the common surface, at which we have ^ = 0', = 0, 
say, and therefore y = jS cos fee, approximately. 

The pressure-equations are 

- = const. —gy—Kil — 2kpe^^ cos fee), 

^. ,, y ...,....(2) 

^ = const, -gy—^il-^- 2A^-*v cos fcc), 
which give, at the common surface, 

^ = const. -(flr-*c«)y,] 

^, r (3) 

S = const. - (flr 4- ic«) y, 

the usual approximations being made. The condition p = p' thus leads to 

c'-l'-^,> w 

a result first obtained by Stokes. 

The presence of the upper fluid has therefore the effect of diminishing 
the velocity of propagation of waves of any given length in the ratio 
{(1 — 8) /{I + «)}*, where 8 is the ratio of the density of the upper to that of 
the lower fluid. This diminution has a two-fold cause ; the potential energy 
of a given deformation of the common surface is diminished in the ratio 
1 — 5, whilst the inertia is increased in the ratio 1 -t- «*. As a numerical 
example, in the case of water over mercury (s"^ = 13*6) the wave- velocity 
is diminished in the ratio '929. 

It is to be noticed, in this and in other problems of the kind, that there 
is a discontinuity of motion at the common surface. The normal velocity 
{dif//dx) is of course continuous, but the tangential velocity (— 9^/8y) changes 
from c (1 — i^ cos kx) to c (1 + ijS cos kx) as we cross the surface; in other 
words we have (Art. 151) a vortex-sheet of strength — 2kcp cos kx. This is 
an extreme illustration of the remark, made in Art. 17, that the free oscil- 
lations of a liquid of variable density are not necessarily irrotational. 

* Thia explains why the natural periods of oscillation of the common surface of two liquids 
of very nearly equal density are yery long compared with those of a free surface of similar extent. 
The fact was noticed by Benjamin Franklin in the case of oil oyer water; see a letter dated 
1762 {CompUU Works, London, n. d., t. ii. p. 142). 

Again, near the mouths of some of the Norwegian fiords there is a layer of fresh over salt 
water. Owing to the comparatively small potential energy involved in a given deformation of the 
common boundary, waves of considerable height in this boundary are easily produced. To this 
cause is ascribed the abnormal resistance occasionally experienced by ships in those waters. See 
Ekman, "On Dead- Water/* Scientific ResuUs of the Norwegian North Polar Expedition, pt. xv. 
Christiania, 1904. 



362 Surface Waves [chap, ix 

If p < p\ the value of c is imaginary. The undisturbed equilibrium- 
arrangement is then unstable. 

2°. The case where the two fluids are confined between rigid horizontal 
planes y = — A, y = A' is almost equally simple. We have, in place of (1), 

Jf , ^ sinh k{y-\-h) , 0' ^sinh k{y- h') , 

(5) 

leadiBgto e* = |.___^_^-£^^^_ (6) 

When kh and kh' are both very great, this reduces to the form (4). When 
Jch' is large, and kh small, we have 

c^=.(l^L)gh, .(7) 

the main effect of the presence of the upper fluid being now the change in 
the potential energy of a given deformation. 

3°. When the upper surface of the upper fluid is free, we may assume 

t-^+z'-A^ir *'-'». ^ (,, 

— = — y + (i3 cosh hy + y sinh ky) cos kx, 

and the conditions to be satisfied at the common boundary, and at the free 
surface, then lead to the equation 

c* (/> coth kh coth kh' + p) - c^p (coth kh' + coth kh) | +(/>-" P') Is = 0. 

(9) 

Since this is a quadratic in c*, there are two possible systems of waves of any 
given length (27r/A). This is as we should expect, for when the wave- 
length is prescribed the system has virtually two degrees of freedom, so that 
there are two independent modes of oscillation about the state of equilibrium. 
For example, in the extreme case where p'/p is small, one mode consists 
mainly in an oscillation of the upper fluid which is almost the same as if 
the lower fluid were solidified, whilst the other mode may be described as an 
oscillation of the lower fluid which is almost the same as if its upper surface 
were free. 

The ratio of the amplitude at the upper to that at the lower surface is 
found to be 

^ . (10) 

kc^ cosh kh' — g sinh kh' 



232-233] Oscillations of Superposed Liquids 363 

Of the yarious special cases that may be considered, the most interesting 
is that in which kh is large ; i.e. the depth of the lower fluid is great compared 
with the wave-length. Putting coth kh = 1, we see that one root of (9) is now 

c» = |, (11) 

exactly as in the case of a single fluid of infinite depth, and that the ratio of 
the amplitudes is e**'. This is merely a particular case of the general result 
stated at the end of Art. 231 ; it will in fact be found on examination that 
there is now no slipping at the common boundary of the two fluids. 

The second root of (9) is, on the same supposition, 

c« _ P-P i^ (12) 

and for this the ratio (10) assumes the value 

-(^-l)6-»*' (13) 

If in (12) and (13) we put M' ^^ oo , we fall back on a former case. If on 
the other hand we make hh' small, we find 

c« = (l - ^') <?A', (14) 

aiid the ratio of the ampUtndes is 



-te-)- 



<p / 

These problems were first investigated by Stokes*. The case of any 
number of superposed strata of different densities has been treated by Webbf 
and Oreenhill;]:. 

233. As a further example of the method of Art. 231 let us suppose that 
two fluids of densities />, />', one beneath the other, are moving parallel to x 
with velocities 17, V\ respectively, the common surface (when undisturbed) 
being of course plane and horizontal. This is virtually a problem of small 
oscillations about a state of steady motion. 

The fluids being supposed unlimited vertically, we assume, for the lower 
fluid 

^=- V{y-^^eo%kcl (1) 

and for the upper fluid 

f « - 17 ' {y - pe-^y cos fee}, (2) 

• "On the Theory of OsciUatory Waves," Camb. Trans, t. viii. (1847) [Papers, t. i. p. 212]. 

t Math. Tripos Papers, 1884. 

t "Wave Motion in Hydrodynamics/' Amer, Jawm, of Maih. i. iz. (1887). 



364 



Surface Waves 



[chap. IX 



the origin being at the mean level of the common surface, which is assumed 
to be stationary-, and to have the form 

y = j8 cos he (3) 

The pressure-equations give 



i-^'^-^-i^'i^-^'-'-^. \ 



P _ 



> = const. -ffy-iU'^(l-\- 2A^e-"»«' cos Jb), 



(*) 



whence, at the common surface, 



?- 



const. 4- (kU^ — g) y. 



2^ = const. - (kU'^ + g) y. 
P 

Since we must have p = p' over this surface, we get 



(5) 



(6) 



This is the condition for stationary waves on the common surface of the 
two currents Uy U\ It may be written 



/ pU + p'U 'y^g p'-p' pp 

\ p-\-p' J k'p-\-p' 






(7) 



The quantity 



pU-hp'U' 
p-h P 



may be called the mean velocity of the two currents; and it appears that 
relatively to this the waves have velocities ± c, given by 






..Au-ur, 



(8) 



where Cq denotes the wave-velocity in the absence of currents (Art. 232). 

If the relative velocity \U — U' \ oi the currents exceed a certain limit, 
given by 

(Cr-Z7')^ = f.^-~f" (9) 

k pp 

the value of c is imaginary, indicating instability. This upper limit diminishes 
indefinitely with the wave-length. 

This result would indicate that, if there were no modifying circumstances, 
the slightest breath of wind would be sufficient to ruffle the surface of water. 
We shall give, later, a more complete investigation of the present problem, 
taking account of capillary forces, which act in the direction of stability. 



233-234] Waves on a Surface of Discontinuity 366 

It appears from (9) that if /> = />', or if j = 0, the plane form of the 
surface is mistable for all wave-lengths. This result illustrates the state- 
ment, as to the instability of surfaces of discontinuity in a liquid, made in 
Art. 79*. 

When the ourrents are confined by fixed horizontal planes y= -h,y=h\we assume 

^=-u[y-p Binhk "^*^}' ^'=-^'{y^P sinhk- ""'M- 

(10) 

The condition for stationary waves on the common surface is then found to be 

pU^ coth kh +p'U'^ ooth kW =| (p -p') (ll)t 

It appears on examination that the undisturbed motion is stable or unstable, according 



as 



^_^.^gpcothfc;i.fp-cothfcft- ^^ ^j2j 

(pp' coth ifcA coth W)* 

where Cq is the wave- velocity in the absence of currents. When h and W both exceed half 
the wave-length, this gives practically the former criterion (9). 

234. These questions of stability are so important that it is worth while 
to give the more direct method of treatment J. 

If <f> be the velocity-potential of a slightly disturbed stream flowing with 
the general velocity V parallel to x, we may write 

<f, = -Vx + <f,, , (1) 

where <fti is small. The pressure-formula is, to the first order, 

M'-"-^''l'+-- <^) 

and the condition to be satisfied at a bounding surface ^ = ^, where t] is small, 
is 

Jt'^^di^"^ ^^^ 

To apply this to the problem stated at the beginning of Art. 233, we 
assume, for the lower and upper fluids, respectively, 

^j = Ce*J'+< (*«-'«, <^/ = (7V*y+»<**"'*> ; (4) 

with, as the equation of the common surface, 

-q = ae«*»-'« (5) 

* The instability was first remarked by Hehnholtz, Lc. ante p. 21. 

t Greenhill, /.e. ante p. 363. 

t Sir W. Tbomaon, " Hydrokinetic Solutions and Observations," Phil Mag, (4), t. zli. (1871) 
[Baltimore Lectures, p. 590]; Rayleigh, "On the Instability of Jets," Proc. Land. Math, 8oc, 
t. X. p. 4 (1878) [Papers, t. i. p. 361]. 



366 Stirface Waves [chap, ix 

The continuity of the pressure at this surface requires, by (2), 

p{i {a -kU)C-j- ga) = p' {% {a -kU')C'-¥ga}\ (6) 

whilst the surface-condition (3) gives 

i(a-kU)a^kC, {{g - kU')a= - kC (7) 

Eliminating a, C, C\ we get 

p(a -kUf -V p' (a -kUy^ gk{p ^ p'), (8) 

whence f = pE±P^^ /{| . PjU^ ^ ^P_ (^7 - V'A. . . . .(9) 
k p-\-p \{kp^p. (p-¥pY^ \r 

leading to the same conclusions as in Art. 233. If 

(U--Uy> ^^'^f^\ \ (10) 

PP 

where Cq is the wave- velocity in the absence of currents, a is imaginary, of 
the form a ± ij3. The complete solution then contains a term with eP* as 
a factor, indicating indefinite increase of amplitude. 

If p = />', it is evident from (8) that a will be imaginary for all values of k. 

Putting U' = — U, we get 

a = ±ikU, (11) 

Hence, taking the real part of (5), we find 

-q = ae*^u* coBkx (12) 

The upper sign gives a system of standing waves whose height continually 
increases with the time, the rate of increase being greater, the shorter the 
wave-length. 

The case of p = p\ with U = U', is of some interest, as illustrating the 
flapping of sails and flags*. We may conveniently simplify the question by 
putting U = U' = 0; any common velocity may be superposed afterwards if 
desired. On these suppositions, the equation (8) reduces to a* = 0. On 
account of the double root the solution has to be completed by the method 
explained in books on Differential Equations. In this way we obtain the 
two independent solutions 

-q = 06***, <^i = 0, <^/ = 0, (13) 

and 7) = ate**^ <^i = - ? e*'' . e**^ <f>i = | e'^y . e**«. . . (14) 

The former solution represents a state of equilibrium; the latter gives a 
system of stationary waves with amplitude increasing proportionally to the 
time. In this form of the problem there is no physical surface of separation 
to begin with ; but if a slight discontinuity of motion be artificially produced, 
e.g. by impulses applied to a thin membrane which is afterwards dissolved, 

* Rayleigh, l.c. 



234-235] Osculations of a HeterogeneoVfS Liquid 367 

the discontinuity will peisist, and, as we have seen, the height of the 
corrugations will continually increase. 

The above method, when applied to the ease where the fluids are oonfined between two 
rigid horizontal planes y= -Ky^h', leads to 

p (<r -itJ7)« coth kh +p' {<r -kUy oath kh'=gk (p -/)» (IS) 

which is equivalent to Art. 233 (11). 

235. The theory of waves in a heterogeneous liquid may be noticed, for 
the sake of comparison with the case of homogeneity. 

The equilibrium value p^ of the density will be a function of the vertical 
co-ordinate (y) only. Hence, writing 

P = Po + p\ /> = />o + />', (1) 

where p^ is the equilibrium pressure, the equations of motion, viz. 

du dp ,dv dp ,^, 

i+«i+'i-»- • w 

, du dp' dv dp' , ... 

become p._ = _^. p„_«_^_^p (4) 

|' + «| = 0' (5) 

small quantities of the second order being omitted. The fluid being incom- 
pressible, the equation of continuity retains the form 

du dv 

ai + ^ = «' (6) 

so that we may write 

" dy' "-dx (^) 

Eliminating p' and p' we find* 

'** - ^/^ if - 4-1} = » 

At a free surface we must have Dp/Dt = 0, or 



(8) 



¥-«|'-^'.| <9> 



Hence, and from (4), we must have 



^-»g OT 



at such a surface. 



* Cf. Love, "Wave Motion in a Heterogeneous Heavy Liquid,'* Proc. Land, Math, Soc, 
t. xxii. p. 307 (1891). 



368 Surf (ice Waves [chap, ix 

To investigate cases of wave-motion we assume that 

X g<«r«-*«) (11) 

The equation (8) becomes 

whilst the surface-condition takes the form 

|-$V-« (>») 

These are satisfied, whatever the vertical distribution of density, by the 
assumption that varies as e^^^ provided 

<^ = i7* (U) 

For a fluid of infinite depth the relation between wave-length and period is 
then the same as in the case of homogeneity (cf. Art. 231), and the motion 
is irrotational. 

For farther investigations it is necessary to make some assumption as to the relation 
between po and y. The simplest is that 

Po«c-^^ (16) 

in which case (12) takes the form 

p-'^-Kt-f)+-»- ('•) 

The solution is 

V^=(^6^'«'+J5e^»') «<<'*-*«>, (17) 

where Xj, X, are the roots of 

X«-pX + (^-l)jk«=0 (18) 

We may apply this to the oscillations of liquid filling a closed rectangular vessel*. 
The quantity h may be any multiple of n-/^, where / denotes the length. If the equations 
to the horizontal boundaries be y =0, y = A, the condition 8^/9a; =0 gives 

^+J5=0, i4c^»*+J5«^=0 (19) 

whence c<^i-^''* = l, or X^ -X,=2t>Tr/A, (20) 

where * is integral. Hence, from (18), 

Xi =i/3+wW^, X2=i/3 -iB7rl\ (21) 



and therefore 



(g-l)i«=X,X.=i/3«+'^* (22) 



We verify that o- is real or imaginary, t.e. the equilibrium arrangement is stable or 
unstable, according as /3 is positive or negative, i.e. according as the density diminishes or 
increases upwardsf. 

* Rayleigh, "Investigation of the Character of the Equilibrium of an Incompressible Heavy 
Liquid of Variable Density," Vroc, Lond. McUh, 8oc. t. adv. p. 170 [Papers, t. ii. p. 200]. 
Reference may also be made to a paper "On Atmospheric Oscillations," Proc. Roy. 8oc. t. Ixzxiv. 
pp. 566, 671 (1910), where another law of density is considered. 

t The case of waves on a liquid of finite depth is discussed by Love (/.c). See also Bumside, 
*'0n the Small Wave-Motions of a Heterogeneous Fluid under Gravity," Proe. Lond, Math. 
Soc. t. xz. p. 392 (1889). 



235-236] Theory of Wave-Groups 369 

236. The investigations of Arts. 227-234 relate to a special type of 
waves ; the profile is simple-harmonic, and the train extends to infinity in 
both directions. But since all our equations are linear (so long as we confine 
ourselves to a first approximation), we can, with the help of Fourier's 
Theorem, build up by superposition a solution which shall represent the 
effect of arbitrary initial conditions. Since the subsequent motion is in 
general made up of systems of waves, of all possible lengths, travelling in 
either direction, each with the velocity proper to its own wave-length, the 
form of the free surface will continually alter. The only exception is when 
the wave-length of every system which is present in sensible amplitude is 
large compared with the depth of the fluid. The velocity of propagation, 
viz. y/{gh\ is then independent of the wave-length, so that in the case of 
waves travelling in one direction only, the wave-profile remains unchanged 
in form as it advances (Art. 170). 

The effect of a local disturbance of the surface, in the case of infinite 
depth, will be considered presently; but it is convenient to introduce in 
the first place the very important conception of * group- velocity,' which has 
application, not only to water-waves, but to every case of wave-motion 
where the velocity of propagation of a simple-harmonic train varies with the 
wave-length. 

It has often been noticed that when an isolated group of waves, of sensibly 
the same length, is advancing over relatively deep water, the velocity of the 
group as a whole is less than that of the individual waves composing it. If 
attention be fixed on a particular wave, it is seen to advance through the 
group, gradually dying out as it approaches the front, whilst its former 
place in the group is occupied in succession by other waves which have come 
forward from the rear *. 

The simplest analytical representation of such a group is obtained by the 
superposition of two systems of waves of the same amplitude, and of nearly 
but not quite the same wave-length. The corresponding equation of the 
free surface will be of the form 

7y = a sin (Jcx — <rf) + a sin (k>x — at) 

= 2a cos{i (h-K)x-\(G- a) t) sin {J (k-^W)x-\(a + a')t}. 

(1) 

If Jk, h' be very nearly equal, the cosine in this expression varies very slowly 
with x\ so that the wave-profile at any instant has the form of a curve of 
sines in which the amplitude alternates gradually between the values and 
2a. The surface therefore presents the appearance of a series of groups of 

* Scott Russell, "Report on Waves," Brit. Aaa. Rep, 1844, p. 369. There is an interesting 
letter on this point from W. Fronde, printed in Stokes* Seientiflc Correspondence, Cambridge, 
1907, t. ii p. 166. 

L. H. 24 



370 Surface Waves [chap, ix 

waves, separated at equal intervals by bands of nearly smooth water. The 
motion of each group is then sensibly independent of the presence of the 
others. Since the distance between the centres of two successive groups is 
27r/(A; — i'), and the time occupied by the system in shifting through this 
space is 27r/((7 — a\ the group- velocity (i7, say) is = (a — ct')/(* — *')» o^" 

"-S (^> 

ultimately. In terms of the wave-length A (= 27r/A;), we have 

where c is the wave- velocity. 

This result holds for any case of waves travelling through a uniform 
medium. In the present application we have 

i 



= (|tanhMr, (4) 



and therefore, for the group- velocity, 

~5F^*K^^Snh2A:A) ^^^ 

The ratio which this bears to the wave- velocity c increases as AA diminishes, 
being \ when the depth is very great, and unity when it is very small, 
compared with the wave-length. 

The above explanation seems to have been first given by Stokes*. The 
extension to a more general type of group was made by Rayleighf and 
Gouyf . The argument of these writers admits of being put very concisely. 
Assuming a disturbance 

y = SC cos (a< - *a? 4- €), (6) 

where the summation (which may of course be replaced by an integration) 
embraces a series of terms in which the values of a, and therefore also of A;, 
vary very slightly, we remark that the phase of the typical term at time 
i-\- ^i and place x-\- Lx differs from the phase at time t and place x by the 
amount uNi — kAx, Hence if the variations of a and k from term to term 
be denoted by 8a and 8k, the change of phase will be sensibly the same for 
all the terms, provided 

&7 A< - SA; Ax = (7) 

* Smith's Piize Examination, 1876 [Papers, t. v. p. 362] . See also Rayleigh, Theory of Sound, 
Art. 191. 

t Nature, t. xxv. p. 62 (1881) [Papers, t. i. p. 640]. 

X "Sur la Vitesse de la lumi^re/' Ann. de Chim, et de Phys. t. xvi. p. 262 (1889). It has 
recently been pointed out that the theory had been to some extent anticipated by Hamilton, 
working from the optical point of view, in 1839; see Havelock, Cambridge Tracts, No. 17 (1914), 
p. 6. 



236] 



Group- Vdocitp 



371 



The group as a whole therefore travels with the velocity 

Ax da 



(8) 



Another derivation of (3) can be given which is, perhaps, more intuitive. 
In a medium such as we are considering, where the wave- velocity varies with 
the frequency, a limited initial disturbance gives rise in general to a wave- 
system in which the different wave-lengths, travelling with different velocities, 
are gradually sorted out (Arts. 238, 239). If we regard the wave-length X 
as a function of x and <, we have 






(9) 



since A does not vary in the neighbourhood of a geometrical point travelling 
with velocity U ; this is, in fact, the definition of U. Again, if we imagine 
another geometrical point to travel with the waves, we have 



3A 3A _ V 3c _ V (fc 3A 
dt dx dx dXdx* 



(10) 



the second member expressing the rate at which two consecutive wave-crests 
are separating from one another. Combining (9) and (10), we are led, again, 
to the formula (3) *. 

This formula admits of a simple geometrical representation f. If a 
curve be constructed with A as abscissa and c as ordinate, the group- 
velocity will be represented by the intercept made by the tangent on the 




♦ See a paper "On Group- Velocity," Proc. Lond, Math. 8oc, (2), t. i. p. 473 (1904). The 
subject is further discussed by G. Green, "On Group- Velocity, and on the Propagation of Waves 
in a Dispersive Medium," Proc. R. 8. Edin. t. ixix. p. 445 (1909). 

t Manch. Mem. t. xliv. No. 6 (1900). 

24—2 



372 SurfcLce Waves [chap, ix 

axis of c. Thus, in the figure, PN represents the wave- velocity for the 
wave-length ON, and OT represents the group- velocity. The frequency of 
vibration, it may be noticed, is represented by the tangent of the angle PON, 

In the case of gravity- waves on deep water, c « A* ; the curve has the 
form of the parabola y* = 4ax, and OT = \PN, i.e., the group- velocity is one- 
half the wave- velocity. 

*237. The group- velocity has moreover a dynamical, as well as a 
geometrical, significance. This was first shewn by Osborne Reynolds*, in 
the ca<se of deep-water waves, by a calculation of the energy propagated 
across a vertical plane. In the case of infinite depth, the velocity-potential 
corresponding to a simple-harmonic train 

f] = a mih (x — d) (11) 

is <f> = ace^^ cos k{x — ct), » (12) 

as may be verified by the consideration that for y = we must have 
37^/3^ = — d<f>/dy. The variable part of the pressure is pd<f>/dt, if we neglect 
terms of the second order, so that the rate at which work is being done on 
the fluid to the right of the plane x is 

- [ P^<iy= pa^k^c^ 8in2 k(x-ct)[ e^^y dy 

= \gpa^c sin* k(x — ct), (13) 

since c* = gjk. The mean value of this expression is \gpa^c. It appears on 
reference to Art. 230 that this is exactly one-half of the energy of the waves 
which cross the plane in question per unit time. Hence in the case of an 
isolated group the supply of energy is sufficient only if the group advance 
with AaZ/the velocity of the individual waves. 

It is readily proved in the same manner that in the case of a finite depth 
h the average energy transmitted per unit time isf 

i^''"''' (i + sJk) (1^) 

which is, by (5), the same as 

\9pa'^^ (15) 

Hence the rate of transmission of energy is equal to the group- velocity, 
d (kc)/dk, found independently by the former line of argument. 

* "On the Rate of Progression of Groups of Waves, and the Rate at which Energy is 
Transmitted by Waves," Nahire, t. xvi. p. 343 (1877) [Papers, t. i. p. 198]. Reynolds also 
constructed a model which exhibits in a very striking manner the distinction between wave- 
velocity and group- velocity in the case of the transverse oscillations of a row of equal pendulums 
whose bobs are connected by a string. 

t Rayleigh, "On Progressive Waves," Proc, Lond, Math. Soc. t. ix. p. 21 (1877) [Papers, 
t. L p. 322]; Theory of Sound, t. i. Appendix. 



23ft-238] Propagation of Energy 373 

This identification of the kinematical group- velocity of the preceding Art. 
with the rate of transmission of energy may be extended to all kinds of waves. 
It follows indeed from the theory of interference groups (p. 369), which is of 
a general character. For let P be the centre of one of these groups, Q that 
of the quiescent region next in advance of P. In a time t which extends over 
a number of periods, but is short compared with the time of transit of a 
group, the centre will have moved to P\ such that PP' = [7t, and the space 
between P and Q will have gained energy to a corresponding amount. 
Another investigation, not involving the notion of * interference,' was given 
by Rayleigh (i.e.). 

Prom a physical point of view the group-velocity is perhaps even more 
important and significant than the wave- velocity. The latter may be greater 
or less than the former, and it is even possible to imagine mechanical media 
in which it would have the opposite direction ; i.e. a disturbance might be 
propagated outwards from a centre in the form of a group, whilst the in- 
dividual waves composing the group were themselves travelUng backwards, 
coming into existence at the front, and dying out as they approach the 
rear*. Moreover, it may be urged that even in the more familiar pheno- 
mena of Acoustics and Optics the wave-velocity is of importance chiefly 
so far as it coincides with the group-velocity. 

238. The theory of the waves produced in deep water by a local 
disturbance of the surface was investigated in two classical memoirs by 
Cauchyf and Poisson;!:. The problem was long regarded as difiBicult, and 
even obscure, but in its two-dimensional form, at all events, it can be pre- 
sented in a comparatively simple aspect. 

It appears from Arts. 40, 41 that the initial state of the fluid is deter- 
minate when we know the form of the boundary, and the boundary-values of 
the normal velocity d<f>/dn, or of the velocity-potential <^. Hence two forms 
of the problem naturally present themselves; we may start with an initial 
elevation of the free surface, without initial velocity, or we may start with 
the surface undisturbed (and therefore horizontal) and an initial distribution 
of surface-impulse {fxf>o). 

If the origin be in the undisturbed surface, and the axis of y be drawn 
vertically upwards, the typical solution for the case of initial rest is 

T) = cos at cos kXy (1) 

(f) = g 6*1' cos kx, (2) 

d 

provided a^ =, gk, (3) 

• Proc. Land. Math. 8oc. (2), t. i. p. 473. 

t l.c. ante p. 16. 

} "M^moire sur )a thdorie dee ondes," Mim, de VAcad. Roy. des Sciences, t. i. (1816). 



374 Surface Waves [chap, ix 

in accordance with the ordinary theory of 'standing' waves of simple- 
harmonic profile (Art. 228). 

If we generalize this by Fourier's double-integral theorem 

/(a;) = - I dk\ /(a) cos k{x — a)da, (4) 

then, corresponding to the initial conditions 

ri^f(x), <^o = 0, (5) 

where the zero suffix indicates surface- value (y = 0), we have 

rj = - I cos atdkl f (a) cos k {x — a) da, (6) 

TT j J -00 

£ f" smat^^^^^ r j.^^^ coBk{x-a)da (7) 

rrj a a J -a> 

I 

If the initial elevation be confined to the immediate neighbourhood of 
the origin, so that/ (a) vanishes for all but infinitesimal values of a, we have, 
assuming 

'" f{a)da=h (8) 



/ 



-00 





This may be expanded in the form 



^ ^ r Sinore ^f:y^^^j^^j^ (9) 



^ = ^r\l ~^* Jfc + ^A:*- . . .1 e^^ cos kxdk, . . . .(10) 

where use is made of (3). If we write 

— y = r cos &, a: = r sin 0, (11) 

we have, y being negative*, 



/ 



n I 
c^^ cos fcx/c"afc = - 





c*v cos kxk^'dk = -^ cos (w + 1) (9, (12) 



so that (10) becomes 

, at (cob d l.-_^«, cos2& . 1 ,- --, cos3d ) ,,^. 

a result which is easily verified. From this the value of t) is obtained by 
Art. 227 (5), putting d = ± lir. Thus, for x > 0, 

^ iTa;|2x 3.5\2x) "^ 3 . 5 . 7 . 9 V2a;/ •••]• "^ '' 

* This formula may be dispensed with. It is sufficient to calculate the value of 4> at points 
on the vertical axis of symmetry ; its value at other points can then be written down at once by a 
property of harmonic functions (cf. Thomson and Tait, Art. 498). 

I That the effect of a concentrated initial elevation of sectional area Q would be of the form 

might have been foreseen from consideration of * dimensions.* 



238] Cauchy-Poisson Wave-Problem 375 

It is evident at once that any particular phase of the surface disturbance, 
e.g., a zero or a maximum or a minimum of t^, is associated with a definite 
value of \gt^jx, and therefore that the phase in question travels over the 
surface with a constant acceleration. The meaning of this somewhat 
remarkable result will appear presently (Art. 240). 

The series in (14) is virtually identical with one (usually designated by 
-M *) which occurs in the theory of FresnePs diffraction-integrals. In its 
present form it is convenient only when we are dealing with the initial stages 
of the disturbance; it converges very slowly when \gt^jx is no longer small. 
An alternative form may, however, be obtained as follows. 

The surface- value of <f> is, by (9), 
<f>o= I cos kxdk 

= -jl sin f Vcft\Ao—\ sin ( (rfj dorj- (15) 

Putting ^ = ;!h©' (^'^ 

wefind l^'sin (— + erf) dcr = ?- r sin (S« - co«) d$, (17) 

J*sin (?-?« erf) d(7= ^ r sin (i« - co2) (iS, (18) 

where "^ = (feT ^^^^ 

Hence <^o = ~ ^f "sin (5« - co^) (?{ (20) 

From this the value of 17 is derived by Art. 227 (5) ; thus 
V = ^ cos ({2 _ co«) dS 

TTX^J 

= ^ jcosco« f "cosC«(i$ + sinco2 Tsin C^d^l (21) 

Trrc* ( ^ / } 

This agrees with a result given by Poisson. The definite integrals are 
practically of Fresnel's forms f, and may be considered as known functions. 

♦ Cf. Rayleigh, Papers, t. iii. p. 129. 
I In terms of a usual aotation we have 

where C (u) = I cob i TU^du, S (ti) = / sin J tu' du, 

the upper limit of integration being u=J(2jt).w. Tables of C(«) and S(ii), computed by 
Gilbert and others, are giyen in most books on Physical Optics. 



376 



Surface Waves 



[OHAP. IX 



Lommel, in his researches on Diffraction*, has given a table of the 
function 



1- 



.2 



+ 



«)«D O«0»i«t7 



(22) 



which is involved in (14), for values of z ranging from to 60. We are thus 
enabled to delineate the first nine or ten waves with great ease. The figure 
below shews the variation of i] with the time, at a particular place; 
for different places the intervals between assigned phases vary as ^/x, whilst 
the corresponding elevations vary inversely as x. The diagrams on p. 377, 
on the other hand, shew the wave-profile at a particular instant ; at different 
times, the horizontal distances between corresponding points vary as the 
square of the time that has elapsed since the beginning of the disturbance, 
whilst corresponding elevations vary inversely as the square of this time. 




[The UDit of the horizontal scale is ^{2x/g). That of the vertical scale is Q/nx, 
if Q be the sectional area of the initially elevated fluid.] 

When gt^l^x is large, we have recourse to the formula (21), which makes 

approximately, as found by Poisson and Cauchy. This is in virtue of the 
known formulae 



Jo Jo 



2V2' 



(24) 



Expressions for the remainder are also given by these writers. Thus 
Poisson obtains, substantially, the semi-convergent expansion 



V = 



9^t 



2*^* 



-♦HS 






-.^{^«-^-^-K^«) 



' /2x 

+ 1.3.5.7.9 (=r 

\gt 



2 i * * * f 



(25) 



* **Bie Bengungserscheinungen geradlinig begrenzter Sohirme," Ahh, d. k. Bayer, Akad, d, 
Wiss, 2^ CI. t. XV. (1886). 



238] 



Waves dtce to a Local JSlevation 



377 




-50-. 



-100' 



0-d 



0.4 



OS 



7 

600i 



400 • 



30a 



20a 



100- • 



-10a 



-200" 



-300. 



-400-^ 




[The unit of the horizontal scales is igt*. That of the vertical scales is 2Q/irgi^.] 



378 Surface Waves [chap, ix 

This is deriyed as follows. We have 

\^y^^'^^ dC= fl e' ^^'-'^ di - J]] e' (^*--"> dC 

= iV^e - ' +_+^-^^^ + ^_^ + ..., (26) 

by a series of partial integrations. Taking the real part, and substituting in the first lino 
of (21), we obtain the formula (25). 

239. In the case of initial impulses applied to the surface, supposed 
horizontal, the typical solution is 

f}^ = cos d e^^ cos kx, (27) 

ri = sin <rf cos kx, (28) 

9P 

with a* = gk as before. Hence, if the initial conditions be 

f4^^F{xl t; = 0, (29) 

1 /•« /■« 
we have <i = — cos a< c*" dk\ F (a) cosk (x — a) da, (30) 

^P.' J -00 

Ti =s I aBmaldk] F (a) cos i (a? — a) ia (31) 

'"'9PJ ^ -00 

For a concentrated impulse acting at the point » = of the surface, we 
have, putting 

r F(a)da=l, (32) 

J —00 

1 f* 
xk =z — I COS att 6*" cos kxdk (33) 

^/>i 

This integral may be treated in the same manner as (9) ; but it is evident 
that the results may be obtained immediately by performing the operation 
l/gp . d/dt upon those of Art. 238. Thus from (13) and (14) we derive 

. 1 (cos^ , -Cos2& 1 ,1 ,«v.cos3& I ,,.. 



^^Trpx^ll 1.3.5V2X/ ■^1.3.5.7.9V2x/ •••!• •• 



(35) 



* 



* With the help of the theory of ' dimenaions * it is easily seen d priori that the effect of a 
concentrated initial impulse P (per unit breadth) is necessarily of the form 

Pt 



23&-239] 



Waves due to a Local Impulse 



379 



The series in (35) is related to the function 



+ 



(36) 



1.3 1.3.5.7 ' 1.3.5.7.9.11 

which has also been tabulated by Lommel. If we denote the series (22) and 
(36) by Si and S,, respectively, we find 

32« 52* 



^ 1.3.5*^1.3.5.7.9 



-,..=J(1 + Si-2jS2), ....(37) 



so that the forms of the first few waves can be traced without difficulty. 

The annexed figure shews the rise and fall of the surface at a particular 
place; for different places the time-iptervals between assigned phases vary 
as \^x, as in the former case, but the corresponding elevations now vary 




[The unit of the horizontal scale is »J{2x/g). That of the yertical scale is 

P /2 

— . / — , where P represents the total initial impulse.] 
npz \ gz 

inversely as x^. In the diagrams on p. 380, which give an instantaneous 
view of the wave-profile, the horizontal distances between corresponding 
points vary as the square of the time, whilst corresponding ordinates vary 
inversely as the cube of the time. 

For large values of \gt^lx, we find, performing the operation Ijgp . djdt 
upon (23), 

'-.fcHS-"-© <'«> 

ap{]froximately. 



Surface Waves 



[oaiP. IX 




7 

1O00O 



[The unit of the horizontal scalee is liTt*. That of the vertical scales is - — r-^. 

The upper curve, if continued to the right, would cross the axis of x and would 
thereafter be indiBtingui^hsble from it on the present scale.} 



23»-240] Interpretation of Results 381 

240. It remains to examine the meaning and the consequences of the 
results above obtained. It wiU be sufficient to consider, chiefly, the case of 
Art. 238, where an initial elevaiion is supposed to be concentrated on a line 
of the surface. 

At any subsequent time t the surface is occupied by a wave-system whose 
advanced portions are delineated on p. 377. For sufficiently small values of 
X the form of the waves is given by (23) ; henoe as we approach the origin 
the waves are found to diminish continually in length, and to increase 
continually in height, in both respects without limit. 

As t increases, the wave-system is stretched out horizontally, proportionally 
to the square of the time, whilst the vertical ordinates are correspondingly 
diminished, in such a way that the area 



i 



ridx 



included between the wave profile, the axis of x, and the ordinates corre- 
sponding to any two assigned phases (i.e., two assigned values of co) is 
constant*. The latter statement may be verified immediately from the 
mere form of (14) or (21). 

The oscillations of level, on the other hand, at any particular place, are 
represented on p. 376. These follow one another more and more rapidly, with 
ever increasing amplitude. For sufficiently great values of t, the course 
of these oscillations is given by (23). 

In the region where this formula holds, at any assigned epoch, the 
changes in length and height from wave to wave are very gradual, so that 
a considerable number of consecutive waves may be represented approxi- 
mately by a curve of sines. The circumstances are, in fact, all approximately 
reproduced when 

Ag = 2«r (39) 

Hence, if we vary t alone, we have, putting At = t, the period of oscillation, 

^=-^5 (*o) 

whilst, if we vary x alone, putting Ax = — A, where A is the wave-length, 
we find 



A = ^ (41) 



* This stfttement does not apply to the case of an initial impuUe, The corresponding pro- 
position then is that 

taken between assigned values of w, is constant. This appears from (34). 



382 Surface Waves [chap, ix 

The wave- velocity is to be found from 

Ag = 0; ., (42) 

thiflgives ^ = T = \/S (*^) 

by (41), as in the case of an infinitely long train of simple-harmonic waves 
of length A. 

We can now see something of a reason why each wave should be con- 
tinually accelerated. The waves in front are longer than those behind, and 
are accordingly moving faster. The consequence is that aU the waves are 
continually being drawn out in length, so that their velocities of propagation 
continually increase as they advance. But the higher the rank of a wave in 
the sequence, the smaller is its acceleration. 

So far, we have been considering the progress of individual waves. But, 
if we fix our attention on a group of waves, characterized as having (approxi- 
mately) a given wave-length A, the position of this group is regulated 
according to (43) by the formula 



-.-iVt-- <") 



i.e., the group advances with a constant velocity equal to hcdf that of the 
component waves. The group does not, however, maintain a constant 
amplitude as it proceeds ; it is easily seen from (23) that for a given value 
of A the amplitude varies inversely as ^/x. 

It appears that the region in the immediate neighbourhood of the origin 
may be regarded as a kind of source, emitting on each side an endless 
succession of waves of continually increasing amplitude and frequency, whose 
subsequent careers are governed by the laws above explained. This persistent 
activity of the source is not paradoxical ; for our assumed initial accumulation 
of a finite volume of elevated fluid on an infinitely narrow base implies an 
unlimited store of energy. 

In any practical case, however, the initial elevation is distributed over 
a band of finite breadth ; we will denote this breadth by I. The disturbance 
at any point P is made up of parts due to the various elements, Sa, say, of 
the breadth I; these are to be calculated by the preceding formulae, and 
integrated over the- breadth of the band. In the result, the mathematical 
infinity and other perplexing peculiarities, which we meet with in the case 
of a concentrated line-source, disappear. It would be easy to write down the 
requisite formulae, but, as they are not very tractable, and contain nothing 
not implied in the preceding statement, they may be passed over. It is 
more instructive to examine, in a general way, how the previous results will 
be modified. 



240] Interpretation of Results 383 

The initial stages of the disturbance at a distance Xy such that Ijx is 
small, will evidently be much the same as on the former hypothesis; the 
parts due to the various elements 8a will simply reinforce one another, and 
the result will be sufficiently expressed by (14) or (23) provided we multiply 

by 



/ 



00 



/ (a) da. 



-00 



i.e., by the sectional area of the initially elevated fluid. The formula (23), 
in particular, will hold when \gt^lx is large, so long as the wave-length A 
at the point considered is large compared with i, i.e., by (41), so long as 
\gt'^lx . Ijx is small. But when, as t increases, the length of the waves at x 
becomes comparable with or smaller than I, the contributions from the 
different parts of I are no longer sensibly in the same phase, and we have 
something analogous to 'interference' in the optical sense. The result 
will, of course, depend on the special character of the initial distribution of 
the values of /(a) over the space Z*, but it is plain that the increase of 
amplitude must at length be arrested, and that ultimately we shall have 
a gradual dying out of the disturbance. 

There is one feature generally characteristic of the later stages which 
must be more particularly adverted to, as it has been the cause of some 
perplexity ; viz. a fluctuation in the amplitude of the waves. This is readily 
accounted for on 'interference' principles. As a sufficient illustration, let 
us suppose that the initial elevation is uniform over the breadth I, and that 
we are considering a stage of the disturbance so late that the value of A in 
the neighbourhood of the point x under consideration has become small 
compared with Z. We shall evidently have a series of groups of waves 
separated by bands of comparatively smooth water, the centres of these bands 
occurring whenever I is an exact multiple of A, say I = nA. Substituting in 
(41), we find 

^-i^J& <*^) 

I 

i.e., the bands in question move forward with a constant velocity, which is, in 
fact, the group-velocity corresponding to the average wave-length in the 
neighbourhood f . 

* Cf. Bumside, **0n Deep-water Waves resulting from a Limited Original Disturbance/* 
Proc. Land, Maih. 8oe, t. zx. p. 22 (1888). 

I This fluctuation was first pointed out by Poisson, in the particular case where the initial 
elevation (or rather depression) has a parabolic outline. 

The preceding investigations have an interest extending beyond the present subject, as 
shewing how widely the effects of a single initial impulse in a dispersive medium (t.e., one in 
which wave-velocity varies with wave-length) may differ from what takes place in the case of 
sound, or in the vibrations of an elastic solid. The above discussion is taken, with some modifica- 
tions, from a paper "On Deep- Water Waves," Proc, Lond. Math. Soe. (2), t. ii p. 371 (1904), 
where also the effect of a local periodic pressure is investigated. 



384 Surface Waves [chap, ix 

The ideal solution of Art. 238 necessarily fails to give any information as to what 
takes place at the origin itself. To illustrate this point in a special case, we may assume 

/(«)=?647- (*«) 

the formula (7) then gives 

A=?^ f^^ «*(»-») 008 fa dft .(47) 

The surface-elevation at the origin is 

ri=^ r COB at e-»dk=^^ T cos crU"^*''' <rda=^ ~ T sin (rte^*^^'' da. . .(48) 
' n Jo ^g Jo irgdtjo 

By a known formula we have* 

1*6-^ Bin 2fizdx=e'-fi*r€?^dx (49) 

Hence, putting »' =gt*/ib, (50) 

we find ''=^<^-*"''*j ^^"^f^"^*"'*'/"^'^) (^^) 

Hence ^(^^"'^^ "S/J^^' ^^2) 

shewing that rjff^ steadily diminishes as t increases. Hence rj can only change sign once. 
The form of the integrals in (48) shews that 17 tends finally to the limit zero ; and it may 
be proved that the leading term in its asymptotic value is - 2QlirgtK 

One noteworthy feature in the above problems is that the disturbance is propagated 
inatantdneonsly to all distances from the origin, however great. Analytically, this might be 
accounted for by the fact that we have to deal with a synthesis of waves of all possible 
lengths, and that for infinite lengths the wave- velocity is infinite. It has been shewn, 
however, by Rayleighf that the instantaneous character is preserved even when the water 
is of finite depth, in which case there is an upper limit to the wave- velocity. The physical 
reason of the peculiarity is that the fluid is treated as incompressible, so that changes of 
pressure are propagated with infinite velocity (cf . Art. 20). When compressibility is taken 
into account a finite, though it may be very short, interval elapses before the disturbance 
manifests itself at any point];. 

241. The space which has been devoted to the above investigation may 
be justified by its historical interest, and by the consideration that it deals 
with one of the few problems of the kind which can be solved completely. 
It was shewn, however, by Kelvin that an approximate representation of 



/; 



* This formula presents itself as a subsidiary result in the process of evaluating 

•00 

e'''*coB2pxdx 


by a contour integration. 

t "On the Instantaneous Propagation of Disturbance in a Dispersive Medium, . . .," PhU, 
Mag. (6), t. xviii. p. 1 (1909) [Papers, t. v. p. 614]. See also Pidduck, "On the Propagation of 
a Disturbance in a Fluid under Gravity," Proc, Boy. Soc. A, t. Ixxxiii. p. 347 (1910). 

X Pidduck, "The Wave-Problem of Cauohy and Poisson for Finite Depth and slightly 
GozDpressible Fluid," Proc, Boy. Soc. A, t. Ixxxvi. p. 396 (1912). 



240-241] Kelvin's Approxirnation 385 

the more interesting features can be obtained by a simpler process, which is 
moreover of very general application*. 

The method depends on the approximate evaluation of integrals of the 
type 

e^f^^^dx (1) 



w = I <j>{x) 

J a 



It is assumed that the circular function goes through a large number of 
periods within the range of integration, whilst <f> (x) changes comparatively 
slowly ; more precisely it is assumed that, when / {x) changes by 27r, <f> (x) 
changes by only a small fraction of itself. Under these conditions the various 
elements of the integral will for the most part cancel by annulling interference, 
except in the neighbourhood of those values of x, if any, for which / {x) is 
stationary. If we write a = a + ^, where a is a value of x, within the range 
of integration, such that/' (a) = 0, we have, for small values of ^, 

/(«)=/(o) + if«Aa). (2) 

approximately. The important part of the integral, corresponding to values 
of a; in the neighbourhood of a, is therefore equal to 



<l> (a) 6<^'«» r e*</'(»)-«*d^, (3) 

J -00 



approximately, since, on account of the fluctuation of the integrand, the 
extension of the limits to ± oo causes no appreciable error. Now by a known 
formula (Art. 238 (24)) we have 

r «*""'^'^^=^-^*=^-«**" w 

Hence (3) becomes 

yrt^?\ ■e«l/c)'^> (5) 



V I i/" (a) 

where the upper or lower sign is to be taken in the exponential according as 
/"(a) is positive or negative. 

If a coincides with one of the limits of integration in (1), the limits in (3) 
will be replaced by and oo , or — oo and 0, and the result (5) is to be halved. 

If the approximation in (2) were continued, the next term would be 
i^V"(a); til© foregoing method is therefore only vaUd under the condition 
that f/'" (a)//" (a) must be small even when ^y ' (a) is a moderate multiple of 
277. This requires that the quotient 

should be small. 

* Sir W. Thomson, "On the Waves produced by a Single Impulse in Water of any Depth, 
or in a Dispersive Medium," Proc. B. S, t. xlii. p. 80 (1887) [PaperSy t. iv. p. 303]. The method 
of treating integrals of the type (1) had however been suggested by Stokes in his paper "On 
the Numerical Calculation of a Class of Definite Integrals and Infinite Seriee," Camb. Trans. 
t. ix. (1850) [Papers, t. ii. p. 341, footnote]. 

L. H. 25 



386 Surface Waves [chap, ix 

In wave-problems of the kind now under consideration, the effect of a 
concentrated initial disturbance is given by formulae of the type 

^ = 1 f *^ (*) e<(-i-*«» ijfc + ^ r^ (jfc) e'^-*+*»» dk (6) 

where a is a known function of k^ viz. 29r/<7 is the period of oedllatioii in a 
train of simple-harmonic waves of length ^fnjk. It is understood that in the 
end only the real part of the expression is to be retained. 

The two terms in (6) represent the results of superposing trains of simple- 
harmonic waves of all possible lengths, travelling in the positive and negative 
directions of a;, respectively. If, taking advantage of the symmetry, we 
confine our attention to the region lying to the right of the origin, the ex- 
ponential in the first integral will alone, as a rule*, admit of a stationary 
value or values, viz. when 

'dk^' <^) 

This detenxiines it as a function of x and t, and we then find, in accordance 
with (5), 

where the ambiguous sign follows that of cPafdk^. The approximation 
postulates the smallness of the ratio 

dV/dP ^ V{M^V^**I*} (9) 

Since 

y (10) 

by (7), it appears that the wave-length and the period in the neighbourhood 
of the point x at time t are 27r/k and 27r/or, respectively. The relation (7) 
shews that the wave-length is such that the corresponding jrow^velocity 
(Art. 236) is x/t. 

The above process, and the result, may be illustrated by various graphical constructionsf. 
The simplest, in some respects, is based on a slight modification of the diagram of Art. 236. 
We construct a curve with X as abscissa and ct as ordinate, where t denotes the time that has 
elapsed since the beginning of the disturbance. To ascertain the nature of the wave- 
system in the neighbourhood of any point z, we measure off a length OQ, equal to z, along 
the axis of ordinates. If PN be the ordinate corresponding to any given abscissa X, the 

* If the group- velocity were negative, as in some of the artificial cases referred to in Art. 237, 
the second integral would be the important one. 

f Proc. of the nth IrUem. Congress of McUhematicians, Cambridge, 1912, p. 281. 



241] 



Geometrical lUustrations 



387 



phaso of the disturbance at x, due to the elementary wave-train whose wave-length is X, 
will be given by the gradient of the Une QP ; for if we draw QB parallel to ON, we have ) 

PR _ PN-OQ _ct-x at-kx 

QB~ ON " X " 2ir 



(11) 




Hence the phase will be stationary if QP be a tangent to the curve ; and the predominant 
wave-lengths at the point x are accordingly given by the abscissae of the points of contact 
of the several tangents which can be drawn from Q, These are characterized by the 
property that the group-velocity has a given value x/t. 

If we imagine the point Q to travel along the straight line on which it lies, we get an 
indication of the distribution of wave-lengths at the instant t for which the curve has 
been constructed. If we wish to follow the changes which take place in time at a given 
point Xf we may either imagine the ordinates to be altered in the ratio of the respective 
times, or we may imagine the point Q to approach in such a way that OQ varies inversely 
as t. 

The foregoing construction has the defect that it gives no indication of the relative 
amplitudes in different parts of the wave-system. For this purpose we may construct the 



art'^kx 




curve which gives the relation between ai as ordinate and k as abscissa. If we draw a 
line through the origin whose gradient is x, the phase due to a particular elementary wave- 
train, viz. at - hzy wiU be represented by the difference of the ordinates of the curve and 

26—2 



388 Surfa4ie Waves [chap, ix 

the straight line. This difference will be stationary when the tangent to the ourve is parallel 
%o the straight line, ».e. when tdafdk =x, as aheady found. It is farther evident that the 
phase-difference, for elementary trains> of slightly different wave-lengths, will vary 
ultimately as the square of the increment of k. Also that the range of values of k for 
which the phase is sensibly the same will be greater, and consequently the resulting 
disturbance will be more intense, the greater the vertical chord of curvature of the curve. 
This explains the occurrence of the quantity td^<r/dJ^ in the denominator of the formula (8). 

In the hydiodjmamical problem of Art. 238 we have* 

^(*)=1, cr««(7*, (12) 

whence 

da/div' i?**"*, d^r/di* = - i^*t"*, d*<Tld]c» = |j*ifc~*. . .(13) 
Hence, from (7), 



k^gt*l^x\ a^gtl2xy (14) 



and therefore 



ah 

V(27r) x^ 



or, on rejecting the imaginary part, 



The quotient in (9) is found to be comparable with {2x/gfl)^, so that the 
approximation holds only for times and places such that ^gt* is large com- 
pared with X. 

These results are in agreement with the more complete iuTestigation of 
Art. 238. The case of Art. 239 can of course be treated in a similar manner. 

It appears from (15), or from the above geometrical construction (the 
curve being now a parabola as in Art. 236), that in the procession of waves 
at any instant the wave-length diminishes continuaDy from front to rear; 
and that the waves which pass any assigned point will have their wave-lengths 
continually diminishing f. 

242. We may next calculate the effect of an arbitrary, but steady, 
application of pressure to the surface of a stream. We shall consider only 
the state of steady motion which, under the influence of dissipative forces^ 

♦ The difficulty as to convergence in this case is met by the remark that the formula (9> 
of Art. 238 gives 

1?=- ^=limj^^Q- / ef^ COB fft 008 kxdk, 

where y is negative before the limit. 

t For further applications reference may be made to Havelock, *'The Propagation of Waves 
in Dispersive Media. . . ,** Proc. Boy. 8oc. t, Ixxxi p. 398 (1908). 



241-242J Surface-Disturhanoe of a Stream 389 

however small, will ultimately establish itself*. The question is in the first 
instance treated directly ; a briefer method of obtaining the principal result 
is explained in Art. 248. 

It is to be noted that in the absence of dissipative forces, the problem is to 
a certain extent indeterminate, for we can always superpose an endless train 
of free waves of arbitrary amplitude, and of wave-length such that their 
velocity relative to the water is equal and opposite to that of the stream, 
in which case they will maintain a fixed position in space. 

To avoid this indet^rminateness, we may avail ourselves of an artifice 
due to Rayleigh, and assume that the deviation of any particle of the fluid 
from the state of uniform flow is resisted by a force proportional to the 
f dative velocity. 

This law of resistance does not profess to be altogether a natural one, 

but it serves to represent in a rough way the eflect of small dissipative forces ; 

and it has the great mathematical convenience that it does not interfere with 

the irrotational character of the motion. For if we write, in the equations of 

Art. 6, 

Z = — /Lt (u — c), Y = — g — fjiv, Z = — fiw, (1) 

where c denotes the velocity of the stream in the direction of x-positive, the 
method of Art. 33, when applied to a closed circuit, gives 



( 



^-f /Lt ] i {udx -{- vdy -{- wdz) == 0, (2) 



whence j{udx -f vdy -f wdz) = Cer*^^ (3) 

Hence the circulation in a circuit moving with the fluid, if once zero, is always 
zero. We now have 

^ = const. -gy^-ii(cx-^<f>)- |j«, (4) 

P 

this being, in fact, the form assumed by Art. 21 (2) when we write 

^ = 9y-lJ^ipX'^4) (5) 

in accordance with (1) above.* 

To calculate, in the first place, the effect of a simple-harmonic distribution 
of pressure we assume 

r = _ a; + pe^y sin Jte, t= -y + pe^y coskx (6) 

c c \ 

* The first steps of the following investigation are adapted from a paper by Rayleigh, **The 
Form of Standing Waves on the Surface of Running Water," Proc, Lond. McUh, Soc t. xv. p. 09 
(1883) [Papers, t. ii. p. 258], being simplified by the omission, for the present, of all reference to 
Capillarity. The definite integrals involved are treated, however, in a somewhat more general^ 
manner, and the discussion of the results necessarily follows a different course. 

The problem had been treated by Popoff, "Solution d'un probl^me sur les ondes permantontes," 
Lumvilh (2), t. iii. p. 251 (1858); his analysis is correct, but regard is not had to the indeter- 
minate character of the problem (in the absence of friction), and the results are consequently 
not pushed to a practical interpretation. 



390 Surface Waves [chap, ix 

The equation (4) becomes, on neglecting the square of A^, 

2 = ... - gy -h j3e*>' (*c* cos ix -f ftc sin kx) (7) 

P 

This gives for the variable part of the pressure at the upper surface (0 « 0) 

Pq = pp{(kc^ — j) cos fee + /Ltc sin fer}, (8) 

which is equal to the real part of 

pp (kc^ — 9 — iy^) «***• 

If we equate the coefficient to 0, we may say that to the pressure 

3>o = C?c«« (9) 

corresponds the surface-form 

9f^-ic-^-iJL,(^^'^' (^«) 

where we have written k for j/c*, so that 2ir/#c is the wave-length of the free 
waves which could maintain their position against the flow of the stream. 
We have also put /li/c = /^i, for shortness. 

Hence, taking the real parts, we And that the surface-pressure 

j>o = Ccos kx (11) 

produces the wave-form 

9^ -<=■ ''- T^.l'if m 

This shews that if /t be small the wave-crests will coincide in position 
with the maxima, and the troughs with the minima, of the applied pressure, 
when the wave-length is less than 27r/#c; whilst the reverse hdds in the 
opposite case. This is in accordance with a general principle. If we impress 
on everything a velocity — c parallel to a;, the result obtained by putting 
/ij = in (12) is seen to be a special case of Art. 168 (14). 

In the critical case of i = #c, we have 

?/>y = - — • sin fer, tl3) 

shewing that the excess of pressure is now on the slopes which face down the 
stream. This explains roughly how a system of progressive waves may be 
maintained against our assumed dissipative forces by a properly adjusted 
distribution of pressure over their slopes. 

243. The solution expressed by (12) may be generalized, in the first 
place by the addition of an arbitrary constant to x, and secondly by a sum- 
mation with respect to i. In this way we may construct the eflect of any 
arbitrary distribution of pressure, say 

Vo=f{x\ (14) 

with the help of Fourier's Theorem (Art. 238 (4)). 



242-243] Surface-Disturbance of a Stream 391 

We will suppose, in the first instance, that / (x) vanishes for all but 
infinitely small values of x, for which it becomes infinite in such a way that 



r f(x)dx^P', (15) 

J -oo 



this will give us the efEect of an integral pressure P concentrated on an 
infinitely narrow band of the surface at the origin. Replacing C in (12) by 
P/tr.Sk, and integrating with respect to k between the limits and oo, 
we obtain 

kP /"* (k — k) cos fee — ui sin fee ,, ,, ^. 

^''y = vi 0—1^3^)*-+^ — ^^ (^^) 

If we put i" = A + »m, where k, m are taken to be the rectangular co-ordinates of a variable 
point in a plane, the properties of the expression (16) are contained in those of the complex 
integral 

(17) 



/i^* 



It is known that the value of this integral, taken round the boundary of any area 
which does not include the singular ppint (C=<^)> ^ zero. In the present case we have 
c = K+ifjLj^, where k. and fi^ are both positive. 

Let us first suppose that x is positive, and let us apply the above theorem to the region 
which is bounded externally by the line m=0 and by an infinite semicircle, described with 
the origin as centre on the side of this line for which m is positive, and internally by 
a small circle surrounding the point (k, fi|). The part of the integral due to the infinite 
semicircle obviously vanishes, and it is easily seen, putting {'-c=re^', that the part due 
to the small circle is 

if the direction of integration be chosen in accordance with the rule of Art. 32. We thus 
obtain 

_«A;-(ic+»/ii) Jo A;-(ic+t^,) 
which is equivalent to 

On the other hand, when x is negative we may take the integral (17) round the contour 
made up of the line m=0 and an infinite semicircle lying on the side for which m is 
negative. This gives the same result as before, with the omission of the term due to the 
singular point, which is now external to the contour. Thus, for x negative, 

Jo k-{K+tfXi) Jo k + {K+%fj^) 

An alternative form of the last term in (18) may be obtained by integrating round the 
contour made up of the negative portion of the axis of k, and the positive portion of 'the 
axis of m, together with an infinite quadrant. We thus find 

/"o gifex /•• e'*"* 

I , 7 r-T dk + / -: ; : — : tdm =0, 

which is equivalent to 

/ j—i ^,dk= — ? r-dm (20) 

Jok+{K+%fA^) Jom~fli+%K 



392 Surface Waves [chap, ix 

This is for x positive. In the case of x negative, we must take as our contour the 
negative portions of the axes of k^ m, and an infinite quadrant. This leads to 

/ 7—7 ^-,dk= r-iw, (21) 

as the transformation of the second member of (19). 

In the foregoing argument fi^ is positive. The corresponding results for the integral 

dC (22) 



/ 



e*^ 



C-(«c-»>i) 

are not required for our immediate purpose, but it will be convenient to state them for 
future reference. For x positive, we find 

/ , / . , dk= -r-7 ^,dk= — = r-dm; (23) 

whilst, for X negative, 

I ^ / . , iifc=-2^te<<^-^^»+ , f . . dk 
Jo *-(«c-»fti) Jo *+(«-»/ 



»>l) 



= -2irie*^-^^' + ^dm (24) 

Jo W -/*! -IK 

The verification is left to the reader*. 



c"» 



If we take the real parts of the formulae (18), (20), and (19), (21), respectively, we 
obtain the results which f oUow. 

The formula (16) is equivalent, for x positive, to 

kP ^ Jo (A: + /c)2 + fjij^ 

==-2.e-^^^sin.x+f (7'-^-);"7t > (25) 

Jo (m-/ti)« + #c2 ^ ' 

and, for x negative, to 

(m + fjLj) e*^* dm 



'3>'i 



(m + fjL^)^+K^ ^^^^ 



The interpretation of these results is simple. The first term of (25) 
represents a train of simple-harmonic waves, on the down-stream side of the 
origin, of wave-length ^nrc^jg, with amplitudes gradually diminishing according 
to the law c""**!*. The remaining part of the deformation of the free-surface, 
expressed by the definite integrals in (25) and (26), though very great for 
small values of x, diminishes very rapidly as x increases in absolute value, 
however small the value of the frictional coefl&cient ^ii. 

When III is infinitesimal, our results take the simpler forms 



-^ . V == -- 27r sin /ex + ,— - — 
kP ^ ./' * + #c 



= — 27r sin #ca; + — r = dm, (27) 



* For another treatment of these integrals, see Dirichlet, YwUjBwngen ueber d. Lehre v. d. 
einfachen u. mehr/achen beatimmten IntegrcUen (ed. Arendt), Braunschweig, 1904, p. 170. 



243] Surfaee-Disturbafice of a Stream 393 

for X positive, and 

-% . y = y— — d* = — r-; — ^ dm, (28) 

for X negative. The part of the disturbance of level which is represented 
by the definite integrals in these expressioniB is now symmetrical with respect 
to the origin, and diminishes constantly as the distance from the origin 
increases. When kx is moderately large we find, by usual methods, the 
semi-convergent expansion 

j w« + fc« K*x* #c*x* "^ #c«a;« ^ ' 

It appears that at a distance of about half a wave-length from the origin, 
on. the down-stream side, the simple-harmonic wave-profile is fully 
established. 

The definite integrals la (27) and (28) can be reduced to known functions as foUows. 
U we put (k + K)x=UfWe have, for x positive, 

Jo «+K J KX tt 

= - Oi o; COS jca; +(^ -Si ica?) sin koj, (30) 

where, in conformity with the usual notation, 

Ciu= - I duj Sitt=/ - — tftt (31) 

;« « Jo u 

The functions Oi u and Si u have been tabulated by Glaisher*. It appears that as u 
increases from zero they tend very rapidly to their asymptotic values and ^tt, respectively. 
For small values of u we have 

It* tt* 



• • • 






U* tt* 



^'«=«-3:3! + 5.5! 



• • • » 



where y is Euler*s constant '5772. . . . 

It is easily found from (25) and (26) that when /l^ is infinitesimal, the 
integral depression of the surface is 



I 



" ydx = ^, (33) 



exactly as if the fluid were at rest. 

* ** Tables of the Nomerical Values of the Sine-Integral, Ck>sine-Integral, and Exponential- 
Integral," Phil, Trans. 1870; abridgments are given by Dale and by Jahnke and Emde. The 
expression of the last integral in (27) in terms of the sine- and cosine-integrals was obtained, in 

a different manner from the above, by Schlomilch, "Sor I'int^ale d^finie / znT* «""**»" Orelle, 

t. xxxiii. (1846); see also De Morgan, Differential and Integral Calculus, London, 1842, p. 654, 
and Dirichlet, Vorlesungen, p. 208. 



394 Surface Waves [chap, ix 

244. The expressions (25), (26) and (27), (28) alike make the elevation 
infinite at the origin, but this difficulty disappears when the pressure, which 
we have supposed concentrated on a mathematical line of the surface, is 
diffused over a band of finite breadth. 

To calculate the effect of a distributed pressure 

Po=/(a') (34) 

it is only necessary to write a; — a for a? in (27) and (28), to replace P by 
/ (a) 8a, and to integrate the resulting value of y with respect to a between 
the proper limits. It follows from known principles of the Integral Calculus 
that if f^ be finite the integrals will be finite for all values of x. 

' In the case of a uniform pressure f^^ applied to the part of the surface 
extending from — oo to the origin, we easily find by integration of (25), for 
a: > 0, 



gpy=-1^,^^^^)^^-^—-^, (35) 

where /n^ ^^ ^^^^ P^^ = ^* Again, if the pressure j>o ^^ applied to the part 
of the surface extending from to + oo , we find, for x < 0,* 

From these results we can easily deduce the requisite formulae for the case 
of a uniform pressure acting on a band of finite breadth. The definite 
integi'al in (35) and (36) can be evaluated in terms of the functions Ciu, 
Si u ; thus in (35) 

r*e-*"*dm r^sinfeCj, ,- «. v r^- - /o^v 

K — =— — i = i dk = liTT — Si Kx) cos #ca; + Ci /ca: sm K*. . . (37) 

In this way the diagram on p. 395 was constructed; it represents the 
case where the band (AB) has a breadth #c~^, or *159 of the length of a 
standing wave. 

The circumstances in any such case might be realized approximately by 
dipping the edge of a slightly inclined board into the surface of a stream, 
except that the pressure on the wetted area of the board would not be uniform, 
but would diimhish from the central parts towards the edges. To secure 
a uniform pressure, the board would have to be curved towards the edges, to 
the shape of the portion of the wave-profile included between the points 
Ay B in the figure. 

It will be noticed that if the breadth of the band be an exact multiple 
of the wave-length (27r//c), we have zero elevation of the surface at a distance, 
on the down-stream as well as on the up-stream side of the source of 
disturbance. 



244] 



Wave-Profile 



395 



396 Surface Waves [chap, ix 

The diagram shews certain peculiarities at the points A^ B due to the 
discontinuity in the applied pressure. A more natural representation of a 
local pressure is obtained if we assume 



y» = L-^ (38) 



We may write this in the form 



p Y p r"^ 
= -.i— !^ = - 6-»+*»«dt, (39) 



n ^ 



provided it is understood that, in the end, only the real part is to be 
retained. On reference to Art. 242 (9), (10) we see that the corresponding 
elevation of the free surface is given by 



= — ,- ^ dk (40) 

n J Q fc — K — tui 



(41) 



gpy = - / — ; .- dm (42) 



By the method of Art. 243, we find that this is equivalent, for 
35 >0, to 

i^«^ = ^ 27rte('+WU-») + J dm 

and, for « < 0, to 

:P f^ g<md+m0 

Hence, taking real parts, and putting /L4 » 0, we find 

o z> ^ft • , P r* m cos m6 — #c sin m6 ^^ , _ ^t 
gpy^- 2KPe-^^ sin #ca? + - j- — ^ «"•*• am, [x > 0], 

(43) 

#cP r*m cos m6 — #c sin m6 ^^ , r ^t 

(44) 

The factor e^* in the first term of (43) shews the effect of diflhising the 
pressure. It is easily proved that the values of y and dy/da> given by these 
formulae agree when a? = 0*. 

245. If in the problem of Art. 242 we suppose the depth to be finite 
and equal to A, there will be, in the absence of dissipation, indeterminateness 
or not, according as the velocity c of the stream is less or greater than (ghy, 
the maximum wave- velocity for the given depth ; see Art. 229. The difficulty 
presented by the former case can be evaded by the introduction of small 
frictional forces; but it may be anticipated from the preceding investigation 
that the main effect of these will be to annul the elevation of the surface 
at a distance on the up-stream side of the region of disturbed pressure, 

* A difFerent treatment of the problem of Art8. 243, 244 is given in a paper by Kelvin, " Deep 
Water Ship- Waves," Proc. R. S. Edin, t. xxv. p. 662 (1906) [Papers, t. iv. p. 368]. 



244-245] Stream of Finite Depth 397 

and if we assume this at the outset we need not complicate our equations 
by retaining the fiictional terms*. • 

For the case of a simple-harmonic distribution of pressure we assume 

? as — x 4- jS cosh lc(y + h) sin fcc, 



JL := — y ^ j8 ginh i (y _|- J) cOg J^x^ 



(1) 



c 
as in Art. 231 (3). Hence, at the surface 

y » j3 sinh M cos Axr, (2) 

we have 

2? = -jy - |(j2 « c«) = jS (*c« cosh *A - J sinh A*) cos ibr, . .(3) 

so that to the imposed pressure 

yo «* cos fcr (4) 

will correspond the surface-form 

_ C sinh tA , 

^ " p' hc^ cosh kh — g sinh kh ^ ' 

As in Art. 242, the pressure is greatest over the troughs, and least over the 
crests, of the waves, or vice versd, according as the wave-length is greater or 
less than that corresponding to the velocity o, in accordance with general 
theory. 



The generalization of (5) by Fourier's method gives 

dk (6) 






I 
^ sinh kh cos kx 



kc* cosh kh-g sinh kh 

as the representation of the effect of a pressure of integral amount P applied to a narrow 
band of the surface at the origin. This may be written 

irpC« _ r COS jXU/h) 

-p-'^-jo uoothu-gh/c^'^'' <^) 

Now consider the complex integral 

r«*» (8) 



/ 



C coth ( - gh/c^ 



where {=:u +iv. The function under the integral sign has a singidar point at ^ = i^ too , 
according as a: is positive or negative, and the remaining singular points are given by the 
roots of 

T-gh ^®' 

Since (6) is an even function of x, it will be sufficient to take the case of x positive. 

* There is no difficulty in so modifying the investigaticn as to take the frictional forces into 
account, when these are very small. 



398 Stirface Waves [chap, ix 

Let us first suppose that c^>gh. The roots of (9) are then all pure imaginaries ; viz. 
they are of the form ±%fi, where 3 is a root of 

tan 3 c* ,,^. 

V^T (^«> 

The smallest positive root of this lies between and ^ir, and the higher roots approximate 
with increasing closeness to the values (a + \) ir, where s is integral. We wiU denote these 
roots in order by ^o> ^» &t» • • • • Let us now take the integral (8) round the contour made 
up of the axis of u, an infinite semicircle on the positive side of this axis, and a series of 

small circles surrounding the singular points C=^^o> i^* ifit 'Hie part due to the 

infinite semicircle obviously vanishes. Again, it is known that if a be a simple root of 
/ (C) =0 the value of the integral 

(C) 



I't 



/(f)* 

taken in the positive direction round a small circle enclosing the point ^=a is equal to* 

2-fg "• (") 

Now in the case of (8) we have 

/ ' (ti) = coth a -a (coth« a - I) =- l^ (l -^^\ + aA (12) 

whence, putting a =ifi„ the expression (11) takes the form 

2»r5.«--^»*'* (13) 

where B,= i ' i (14) 

The theorem in question then gives 

/ ^ r7-.<^^+ 4 r7-a<i^-2»rS.5.c-^'«'*=0 (16) 

J ^^uoothu-gh/c* J qU coth u-gh/c* o • 

If in the former integral we write - u for u, this becomes 

r_£2i(25/*) rf„=,S:B.«-^'.«/» (16) 

J ouoothu-gh/c^ ^ ' ^ ' 

The surface-form is then given by 

y=^..sr*.e-'-'* (17) 

It appears that the surface-elevation (which is symmetrical with respect to the origin) 
is insensible beyond a certain distance from the seat of disturbance. 

When, on the other hand, c^<gh, the equation (9) has a pair of real roots ( ±a, say), the 
lowest roots ( ±/3o) of ( 10) having now disappeared. The integral (7) is then indeterminate, 
owing to the function under the integral sign becoming infinite within the range of 
integration. One of its values, viz. the * principal value,' in Cauchy's sense, can however 
be found by the same method as before, provided we exclude the points f = ±a from the 
contour by drawing semicircles of small radius € round them, on the side for which v is 
positive. The parts of the complex integral (8) due to these semicircles will be 

* Forsyth, Theory of Functions, Art 24. 



245-246] Stream of Finite Depth 399 

where /^ (a) is given by (12) ; and their sum is therefore equal to 

2irA sin "^ (18) 

« • • • 

where A- i , \ (19) 

, gh (gh \ 

The equation corresponding to (16) now takes the form 

SO that, if we take the principal value of the integral in (7), th^ surface-form on the side 
of X positive is 

y=-— ,^8in^ + ^,Sr5.e-^«'*. (21) 

Hence at a distance from the origin the deformation of the surface consists of the 
simple-harmonic train of waves indicated by- the first term, the wave-length 2irh/a being 
that corresponding to a velocity of propagation c relative to still water. 

Since the function (7) is symmetrical with respect to the origin, the corresponding 
result for negative values of rr is 

y=^.^sinf +|,S>.e«."» (22) 

The general solution of our indeterminate' problem is completed by adding to (21) and 
(22) terms of the form 

CcoB -T-+Dsm-T- (23) 

The practical solution, including the effect of infinitely small dissipative forces, is obtained 
by so adjusting these terms as to make the deformation of the surface insensible at 
a distance on the up-stream side. We thus get, finally, for positive values of x. 



y^-^^A^f^^XB.e-^-^ (24) 



and, for negative values of z. 



y=|iS«" *.«"•"'* (26) 



For a different method of reducing the definite integral in this problem we must refer 
to the paper by Kelvin cited below. 

246. The same method can be employed to investigate the efiect on a 
uniform stream of slight inequalities in the bed*. 

Thus, in the case of a simple-harmonic corrugation given by 

y = — h + y cos kx, (1) 

♦ Sir W. Thomson, "On Stationary Waves in Flowing Water,'* Phil, Mag. (6), t. xxu. pp. 353, 
445, 517 (1886), and t. xxiii. p. 52 (1887) [Papers, t."iv; p. 270]. 



400 Surface Waves [chap, ix 

the origin being as usual in the undisturbed surface, we assume 

? = — a: 4- (a cosh hy -\- P sinh hy) sin fee, 

' y (2) 

1? = — y -f (a sinh hy •\- P cosh hy) cos fee. 
c 

The condition that (1) should be a stream-line is 

y = — a sinh kh-{- p cosh kh (3) 

The pressure-formula is 

2 = const. — ^ + A»* (a cosh hy -{- fi sinh hy) cos fcr, (4) 

P 

approximately, and therefore along the stream-line ^ = 

^ = const, -f {kc^a — gP) cos fcr, 

so that the condition for a free surface gives 

jfcc*a - jjS = (5) 

The equations (3) and (5) determine a and j3. The profile of the free surface 
is given by 

If the velocity of the stream be less than that of waves in still water 
of uniform depth h, of the same length as the corrugations, as determined by 
Art. 229 (4), the denominator is negative, so that the undulations of the free 
surface are inverted relatively to those of the bed. In the opposite case, the 
undulations of the surface follow those of the bed, but with a difierent vertical 
scale. When c has precisely the value given by Art. 229 (4), the solution 
fails, as we should expect, through the vanishing of the denominator. To 
obtain an intelligible result in this case we should be compelled to take special 
account of dissipative forces. 

The above solution may be generalized, by Fourier's Theorem, so as to apply to the 
case where the inequalities of the bed foUow any arbitrary law. Thus, if the profile of 
the bed be given by 

y=-h+f{x)=-k+- rdkT /(f) COS fc (a: -f) if, (7) 

n J J _oo 

that of the free surface will be obtained by superposition of terms of the type (6) due to 
the vajriouB elements of the Fourier-integral; thus 

f{^)coek{x-i) ^ g. 

cosh kh - g/kc^ . sinh kh 

In the case of a single isolated inequality at the point of the bed vertically beneath 
the origin, this reduces to 

_G r* cos fee „ 

n ] cosh kh -g/kc^ . sinh kh 

u cos (xu/h) 



=- r «^* r 

^ J J — 00 



vh J U 



cosh u - gh/c* . sinh u 



du, (9) 



246-247] Inequalities in the Bed of a Stream 401 

where Q represents the area included by the profile of the inequality above the general 
level of the bed. For a depression Q will of course be negative. 

The discussion of the integral 



/ 



f cosh f - gh/c^ , sinh f 



can be conducted exactly as in Art. 245. The function to be integrated differs only by 
the factor ^/(sinh {) ; the singular points therefore are the same as before, and we can at 
once write down the results. 

Thus when c^>gh we find, for the surface-form, 

the upper or the lower sign being taken according as 2; is positive or negative. 
When c^<ghy the * practical' solution is, for x positive, 

j,=_^4^^8Hif +«srs.^«-'''«'» (12) 

^ h sinha ^ A 1 'smjy, ^ ' 

and, for X negative, ^=1 ^1" ^'SS^ *^**'* (^^^ 

The symbols a, /3«, A^ B^ have here exactly the same meanings as in Art. 246*. 

247. We may calculate, in a somewbat similar manner, the disturbance 
produced in the flow of a uniform stream by a submerged cylindrical obstacle 
whose radius h is small compared with the depth/ of its axisf. The cylinder 
is supposed placed horizontally athwart the stream. 

We write 

^ = -«c(l+^^)+X, (1) 

where c denotes as before the general velocity of the stream, and r denotes 
distance from the axis of the cylinder, viz. 

r = V{a;* + (y+/)'} (2) 

the origin being in the undisturbed level of the surface, vertically above the 
axis. This makes 3<^/3r = f or r = 6, provided x ^® negligible in the neigh- 
bourhood of the cylinder. 

We assume 

/•OO 

X = I a{k) e*«' sin kcdh, (3) 

J 

* A very interesting drawing of the wave-profile produced by an isolated inequality in the bed 
is given in Kelvin's paper, Phil. Mag. (5), t. xxii. p. 520 [Paperst t. iv. p. 296]. The case of an 
abrupt change of level in the bed is discussed by Wien, Hydrodynamikf p. 201. The effect of 
inequalities of various kinds has been investigated by Qsotti in recent volumes of the RcTid. 
della r. Accad, dei Lincei, on the supposition that c' is so large in comparison with gh that the 
infiuence of gravity may be neglected. The problems are accordingly of the type considered in 
Arts. 73 . . . supra. 

t The investigation is taken from a paper "On some cases of Wave-Motion on Deep Water," 
Ann. di matematica (3). t. xxi. p. 237 (1913). I find that the problem had been suggested by 
Kelvin, Papers, t. iv. p. 369 (1904). 

L. H. 26 



402 Surface Waves [chap, ix 

where a (A;) is a function of i, to be determined. For the equation of the 
free surface, assumed to be steady, we put 

7] = I j3(i) cos kxdk (4) 

J 

The geometrical condition to be satisfied at the free surface is 

dy ""dx' <^^ 

wherein we may put y = 0. Since (1) is equivalent to 

^ = - c» - 6«c e-*<>'+/' sin kxdk 4- x» (6) 

J 

for positive values of y +/, this condition is satisfied if 

ft2ce-»/ 4- a(k) = cj3(Jfe) (7) 

Again, the variable part of the pressure at the free surface is given by 

?-»-4(l)' 



• 00 

= — jiy — |c* — 6*c* I c~*' cos kxkdk + c 

J 



^ 

dx 



/•oo roo 

= — jiy — ^c* — 6*c* / e-*' cos kxkdk -f- c 1 a (A) cos kxkdk, (8) 

where terms of the second order in the disturbance have been omitted. This 
expression will be independent of x provided 

gp(k) 4- Jfc6*c«e-*^ - kca{k) = (9) 

Combined with (7), this gives 

a(*) = |i^6«ce-*/, ^(A) = 2^^. (10) 



where K = g/c^ (11) 

as in Art. 242. Hence 

= 26» I 

J I 



* fe-*^ COS fcrd* 



^f. * "" jl^-^*' <'2> 



The integral is indeterminate, but if a; be positive its principal value is equal 
to the real part of the expression 

/« p—imf—mx 
-.- dm (13) 
tm — K 



247-248] Waves dtte to a Submerged Cplinder 403 

Adopting this we have 

71 = ,- -J,!, — 27ric6*e"*^ sin kx 
' «*+/* 

o r« /"" C*^ sin wif — m cos m/) e-*»* , ,. .. 

For large values of x the second term is alone sensible. 

Since the value of 77 in (12) is an even function of x we must have, for 
X negative, 

26*/ « !• * . « ,0 r*(Ksinm/— mcosm/*)e'** , ,--. 

17 = -5—=^ + ^Kb^e"^ sm ica; - 2#c62 ^ =^—5- — ^-^^ dm. ... (15) 

' x^-\'P Jo m* + ic* 

On the disturbances represented by these formulae we can superpose any 
system of stationary waves of length 27r/#c, since these could maintain their 
position in space, in spite of the motion of the stream ; and if we choose as 
our additional system 

7) = — 27^06*6"*^ sin #cx (16) 

we shall annul the disturbance at a distance on the up-stream side (x < 0), 
as is required for a physical solution. The result is 

26*/ \ 

17 = g *^^g — 4jr#c6*e"*^ sin kx -f- &c. [x > 0], 

^ ^^ [ (17) 

^ = a;2 ^ ya -^ ^^- [a;<0]. 

It appears that there is a local disturbance immediately above the obstacle, 
followed by a train of waves of length 27rc^/g on the down-stream side*. 

248. If in the problems of Arts. 243, 245 we impress on everything a 
velocity — c parallel to x, we get the case of a pressure-disturbance advancing 
with constant velocity c over the surface of otherwise still water. In this 
form of the question it is not difficult to understand, in a general way, the 
origin of the train of waves following the disturbance. 

If, for example, equal infinitesimal impulses be applied in succession to 
a series of infinitely close equidistant parallel lines of the surface, at equal 
intervals of time, each impulse will produce on its own account a system of 
waves of the character investigated in Art. 239. The systems due to the 
different impulses will be superposed, with the obvious result that the only 
parts which reinforce one another will be those which have the wave-length 
appropriate to the velocity c with which the disturbing influence advances 
over the surface, and which are (moreover) travelling in the direction of this 
advance. And the investigations of Arts. 236, 237 shew that the groups of 

* If we investigate the asymptotic expansion of the definite integral in (13), when k/ is 
large, we find on substitution in (12) that the most important term gives -2b'^/l{^+P), and so 
cancels the first term in the above values of ri. 

2ft— 2 



404 



Surface Waves 



[OHAP. IX 



waves, of this particular length, which are produced, are continually being 
left behind. 

This question can be treated from a general standpoint by an application 
of Kelvin's method (Art. 241). 

Let us suppose that the disturbing influence, imagined to be concentrated 
in a line perpendicular to the axis of x, is advancing with the constant velocity 
c in the direction of x negative. Let be its position at the instant under 
consideration, and let t denote the time that has elapsed since it occupied 



Q 



any previous position Q, so that OQ = ct. It is required to find the actual 
disturbance at any assigned point P. 

We write 

X = OPy $ = PQ = ct — X (1) 

Taking the formula (8) of Art. 241 as a basis, we have for that part of the 
disturbance at P which was originated at Q an expression of the type 

where k is now to be regarded as a function of t determined by the relation 

tp^^i=^ct-x, (3) 

and m = V i \td^<Tldk^ \ (4) 

The relation between a and k will depend of course on the dynamics of the 
particular kind of waves we are considering. 

We have to integrate (2) with respect to t between the limits and oo , 
but only those elements of the integral are important for which the exponential 
is nearly stationary in value. Now, having regard to (3), 



248] Waves diie to a Travelling Disturbance 405 

whilst by differentiation of (3) 

d^a dk da ___ ^ 
^Wdi'^dk^^"^' 

Hence, writing V for the group- velocity {da/dk), we have 

^ (of - if) = cr - Ac, ^, (of - k$) = ^ ^ ^^' , (5) 

where the sign is the opposite to that of d^a/dk*. 

In order that the phase at P, viz. at — k^± Jtt, may be stationary as 
regards variation in the position of Q we must have, then, 

a=kc (6) 

This determines k, and the corresponding value of t then follows from (3). 
If we denote these special values of k and t by k and r, respectively, the 
corresponding value of at — k^ will be 

ar — K^ = k{ct — i)= KX, 
by (6). Hence if we write 

k^K-hk', t = T + t\ (7) 

the index in (2) will take the form 

approximately. Since 

" c+»«(«^-<'W'«'(fe' = -'^T''^, (8) 

-flo \U — c\ 

by Art. 241 (4), we obtain finally, for the disturbance at the point x, the 
simple expression 

>? = |^L^e- (9) 

It will be understood that U here denotes the value of the group-velocity 
corresponding to k = k. 

It appears from (6) that the wave-length of the progressive wave-train 
represented by this formula is that of a free wave-train whose velocity of 
propagation is c. Also, since by (3) 

x = {C'-'U)t, (10) 

the values of a; to which the preceding calculation applies will be positive or 
negative according as J7 $ c. If ?7 < c, as is the case of gravity waves on 
a liquid, the train follows the initiating disturbance, whilst if 17 > c, as is the 
case of capillary waves (Art. 266), it precedes it. 

If there is more than one value of k satisfying (6) there will be a term in 
7) corresponding to each of these. 



/ 



406 Surface Waves [chap, ix 

Referring now to Art. 239 (33), we see that to find the elevation 17 in the 
case of waves on deep water due to a travelling pressure we must put 

<f> (k) ^ iaP/gp (11) 

Since U is now = Jc, we obtain, on taking the real part, 

77 = sinKX. (12) 

9P 

in agreement with (27) of Art. 243*. 

As the preceding investigation involves a double approximation, it may 
be worth while to give another method of arriving at the result (9) which will 
indicate very readily the condition under which it holds. 

If we introduce the hypothesis of a small frictional force varying as the 
velocity, the formula (6) of Art. 241, when modified so as to apply to the 
case of a travelling disturbance, takes the form 

TJ = ~r j r<^ (k) 6<<r*-i*(ct-a:) dJc ^ r ^Urt+mct^a^) dk\ 6'^^^ dt. . . (13) 

The integration with respect to t gives 

"^ "" 27rj i/i - t (a-~kc) ^27tJ J/i -> i (a -h kc) ^^^^ 

The quantity /x is by hypothesis small, and will in the limit be made to vanish. 
The most important part of the result will therefore be due to values of k 
in the first integral which make 

a=kc (15) 

approximately. Writing as before k = k-{- k\ where #c is a root of this 
equation, we have 

--kc=(^-c)k'=={U-c)kf, (16) 

nearly, where U denotes the group- velocity corresponding to the wave-length 
2n/K. The important part of (14) is therefore 

Now if a be positive we havet 

p e<"»* dm ^ f27re-««, [x > 0] 

J^aoa + im [ 0, [a;<0] ^ ' 

* This method of obtaining the formula (12) was indicated in a footnote on p. 416 of the 
preceding edition. 

t The results quoted are equivalent to the familiar formulae 

mxdin 



a 



/"* cos mxdm _ . f* m sin w 



wi* 



=ire*«« 



(where the upper or lower sign is to be chosen according as a; is positive or negative), but can be 
obtained directly by a contour integration. 



248-249] Waves dtie to a Travelling Disturbance 407 



whilst 

Hence ii U < c 



p e*"^ dm _ 
J ^oo a — im 



0, [X > 0] 
27re«*. [a;<0] ^ ^ 



^^ <f>(K) e^' ^,|^;^(„C7)^ or 0, (20) 



C-I7 
according as a; % ; whilst it U > c 

^ = 0, or ^M^%-i^«/(cr-e) (21) 

' U — c 

in the respective cases. If we now make /i -^ we obtain our former results. 

The approximation in (16) is valid only if the quotient 

d^a/dk^ .k' -^(U-c) (22) 

is small even when k'x is a moderate multiple of 27r. This requires that 

d^a/dk^ H- (£7 - c) a; (23) 

should be small. Unless V = c, exactly, the condition is always fulfilled if 
X be sufficiently great. It may be added that the results (20), (21) are accurate, 
in the sense that they give the leading term in the evaluation of (14) by 
Cauchy's method of residues. Cf. Art. 242. 

249. The preceding results have a bearing on the question of 'wave- 
resistance.' Taking for definiteness the case of Z7 < c, let us imagine a fixed 
vertical plane to be drawn in the rear of the disturbing agency. If E be the 
mean energy of the waves, the space in front of this plane gains, per unit 
time, the additional energy cE, whilst the energy transmitted across the 
plane is VE, by Art. 237. Hence if R be the resistance experienced by 
the disturbing body 

r^^hIe (1) 

Ii U > Cf the fixed plane must be taken in advance, and the result is 

R = ^^::-^E (2) 

Thus, in the case of a disturbance advancing with velocity c [< ^/{gh)'\ 
over still water of depth A, we find, on reference to Art. 237, 



R 



= *"«-(■ -iJir*)- (') 



where a is the amplitude of the waves. As c increases from to ^/(gh), kK 
diminishes from oo to 0, so that R diminishes from \gpa^ to 0. When 
c>\/(gh\ the effect is merely local, and /? = 0*. It must be remarked, 

♦ Cf. Sir W. Thomson, "On Ship Waves," Proc. Inst. Mech. Eng. Aug. 3, 1887 [Popular 
Lectures and Addresses, London, 1889-94, t. iii. p. 450]. A foimula equivalent to (3) was given 
in a paper by the same author, Phil. Mag. (5), t. xxii. p. 451 [Pajfersj t. iv. p. 279]. 



408 Surfdce Waves [chap, ix 

however, that the amplitude a due to a disturbance of given type will also 
vary with c. For instance, in the case of Art. 244 (43), a oc Ke-^^y where 
K = ^/c*, the depth being infinite. Hence 

R a c-*e-*»^/«' (4) 

An interesting variation of the general question is presented when we have 
a layer of one fluid on the top of another of somewhat greater density. If 
p, p' be the densities of the lower and upper fluids, respectively, and if the 
depth of the upper layer be A', whilst that of the lower fluid is practically 
infinite, the results of Stokes quoted in Art. 232 shew that two wave-systems 
may be generated, whose lengths {^Jk) are related to the velocity c of the 
disturbance by the formulae 

k' ^ " pcothKh' + p''k ^^^ 

It is easily proved that the value of k determined by the second equation is 
real only if 

c2<P:^gh^ (6) 

P 

If c exceeds the critical value thus indicated, only one type of waves 
will be generated, and if the difference of densities be slight the resistance 
will be practically the same as in the case of a single fluid. But if c fall 
below the critical value, a second type of waves may be produced, in which 
the amplitude at the common boundary greatly exceeds that at the upper 
surface; and it is to these waves that the * dead-water resistance' referred to 
in Art. 232 is attributed*. 

The problem of the submerged cylinder (Art. 247) furnishes an instance 
where the wave-resistance to the motion of a solid can be calculated. The 
mean energy, per unit area of the water surface, of the waves represented by 
the second term in equation (14) of that Art. is 

E = ^p {i^KbH-^f)^. 

Since U = Jc, we have from (1) 

R = 4^^gpb^K^e-^f (7) 

For a given depth (/) of immersion, this is greatest when k/= 1, or 

c = V{9f) (8) 

In terms of the velocity c we have 

R = i^^g^ph* . c-«e-«'^/^' (9) 

The graph of 12 as a function of c is appended!. 

* Ekman, l.c. ante p. 361. 

t Ann, di mat.. I.e. The same law of resistance as a function of the velocity c has heen 
obtained by Havelock, in the case of various types of surface disturbance, "Ship Resistance. . .." 
Proc. R. 8. t. Ixxziz. p. 489 (1913). A previous paper by the same author on "The Wave-Making 
Resistance of Ships, . . .," Proc. R. S. t. Ixxxii. p. 276 (1909), may also be referred to. 



249-250] 



Wave-Hesistance 



409 




^fgf) 



Waves of Finite Amplitude. 

250. The restriction to "infinitely small' motions, in the investigations 
of Arts. 227, . . . implies that the ratio (a/A) of the maximum elevation to the 
wave-length must be small. The determination of the wave-forms which 
satisfy the conditions of uniform propagation without change of type, when 
this restriction is abandoned, forms the subject of a classical research by 
Stokes*. 

The problem is most conveniently treated as one of steady motion. If 
we neglect small quantities of the order a'/A*, the solution of the problem in 
the case of infinite depth is contained in the formulae I 



- = — X -f j3e*^ sin fee. 



?- r= — y + )3e*v cos kx (1) 



The equation of the wave-profile = is found by successive approxi- 
mations to be 

y = j3e*v cos A:x = j3 (1 + % + \k^y^ + . . . ) cos fee 
= i*^2 + jS (1 + gPjS2) cos kx + \kp^ cos2fer + p«j3» cosSfec + . . . ; . .(2) 

or, if we put jS (1 + %h^p^) = a, 

y — \ka^ = a cos kx + \ka^ cos 2fei; + fi^a' cos Sfec + (3) 

* "On the theory of Osoillatoiy Waves," CavA, Trans, t. yiii. (1847) [reprinted, with a 
"Supplement," Papers, t. i. pp. 197, 314]. 

The outlines of a more general investigation, including the case of permanent waves on the 
common surface of two horizontal currents, have been given by Helmholtz, *'Zur Theorie von 
Wind und Wellen," Bed. MonaUber. July 25, 1889 [Wiss. Ahh. t. iii. p. 309]. See abo Wien, 
Hydrodynamik, p. 169. 

t Rayleigh, l.c, ante p. 252. 



410 Surface Waves [chap, ix 

So far as we have developed it, this coincides with the equation of a trochoid, 
in which the circumference of the rolhng circle is %v\hy or A, and the length 
of the arm of the tracing point is a. 

We have still to shew that the condition of uniform pressure along this 
stream-Une can be satisfied by a suitably chosen value of c. We have, from 
(1), without approximation 

^ = const, -gy- \c^ {1 - 2ij86*»' cos ht + PjS^e**''}, (4) 

and therefore, at points of the line y — jSe*" cos fee, 

^ = const. ^{h?-g)y- p^c^jS^e^kv 

= const. + (*c* - 5^ - Pc2j8«) y + (5) 

Hence the condition for a free surface is satisfied, to the present order of 
approximation, provided 

c2 = 1+ Pc2j82 = |(1 + A:«a2) (6) 

This determines the velocity of progressive waves of permanent type, and 
shews that it increases somewhat with the amplitude a. 

For methods of proceeding to a higher approximation, and for the 
treatment of the case of finite depth, we must refer to the original in- 
vestigations of Stokes*. 

The figure shews the wave-profile, as given by (3), in the case of fci = J, 
or a/A = -0796. 



The approximately trochoidal form gives an outUne which is sharper near 
the crests, and flatter in the troughs, than in the case of the simple-harmonic 
waves of infinitely small amplitude investigated in Art. 229, and these 
features become accentuated as the amplitude is increased. If the trochoidal 
form were exact, instead of merely approximate, the Umiting form would 
have cusps at the crests, as in the case of Gerstner's waves to be considered 
presently. 

In the actual problem, which is one of irrotational motion, the extreme 
form has been shewn by Stokes f, in a very simple manner, to have sharp 

* See also Rayleigh, PhU, Mag. (6), t. xxi. p. 183 (1911), and Pwc, Boy. 8oc. A, t. xci. p. 345 
(1915). The latter paper includes the case of standing waves. Reference may be made also 
to Priestley, Camb. Proc. t. xv. p. 297 (1909), and Wilton, PhU. Mag. (6), t. xxvii. p. 385 (1914). 

t Papers, t. i. p. 227. 



250] Finite Waves of Permanent Type 411 

angles of 120°. The question being still treated as one of steady motion, 
the motion near the angle will be given by the formulae of Art. 63 ; viz. if 
we introduce polar co-ordinates r, with the crest as origin, and the initial 
line of 6 drawn vertically downwards, we have 

= Of" cos md, (7) 

with the condition that = when fl = ± a (say), so that ma = Jtt. This 
formula leads to 

g = wCr«-i, (8) 

where q is the resultant fluid-velocity. But since the velocity vanishes at 
the crest, its value at a neighbouring point of the free surface will be given by 

q^ = 2gr cos a, (9) 

as in Art. 24 (2). Comparing (8) and (9), we see that we must have m = f , 
and therefore a = Jtt*. 

In the case of progressive waves advancing over still water, the particles 
at the crests, when these have their extreme forms, are moving forwards with 
exactly the velocity of the wave. 

Another point of interest in connection with these waves of permanent 
type is that they possess, relatively to the undisturbed water, a certain 
momentum in the direction of wave-propagation. The momentum, per wave- 
length, of the fluid contained between the free surface and a depth h (beneath 
the level of the origin) which we will suppose to be great compared with A, is 

- p I p^ dxdy = pchX, (10) 

since ^ = 0, by hypothesis, at the surface, and = ch, by (1), at the great depth 
h. In the absence of waves, the equation to the upper surface would be 
y = JA»^ by (3), and the corresponding value of the momentum would 
therefore be 

pc(A + iJfca»)A (11) 

The difference of these results is equal to 

Trpa^Cy (12) 

which gives therefore the momentum, per wave-length, of a system of 
progressive waves of permanent type, moving over water which is at rest at 
a great depth. 

* The wave-profile has been investigated and traced by Michell, "The Highest Waves in 
Water," Phil, Mag. (5), t. xxxvi. p. 430 (1893). He finds that the extreme height is -142 X, and 
that the wave- velocity is greater than in the case of infinitely small height in the ratio of 1*2 to 1. 
See also Wilton, PhU, Mag. (6), t. zxvi p. 1053 (1913). 



412 Surface Waves [chap, ix 

To find the vertical distribution of this momentum, we remark that the 
equation of a stream-line ^ = cV is found from (2) by writing y '\-h' for y, 
and jSe"-**' for j3. The mean-level of this stream-line is therefore given by 

y = - A' -I- J*^%-»*' (13) 

Hence the momentum, in the case of undisturbed flow, of the stratum of 
fluid included between the surface and the stream-line in question would 
be, per wave-length, 

/>cA{A' + iA^«(l-«-"^)} (14) 

The actual momentum 1>eing pch'\ we have, for the momentum of the same 
stratum in the case of waves advancing over still water, 

-npa^c (1 - e-»*') (15) 

It appears therefore that the motion of the individual particles, in these 
progressive waves of permanent tjrpe, is not purely oscillatory, and that there 
is, on the whole, a slow but continued advance in the direction of wave- 
propagation*. The rate of this flow at a depth K is found approximately by 
differentiating (15) with respect to h\ and dividing by pA, viz. it is 

Jfc«a«ce-»*' (16) 

This diminishes rapidly from the surface downwards. 

251. A system of eocfui, equations, expressing a possible form of wave- 
motion when the depth of the fluid is infinite, was given so long ago as 1802 
by Gerstnerf, and at a later period independentiy by Bankine]:. The 
circumstance, however, that the motion in these waves is not irrotational 
detracts somewhat from the physical interest of the results. 

If the axis of x be horizontal, and that of y be drawn vertically upwards, 
the formulae in question may be written 

a; = a 4- T e** sin i (a -I- ci), y = 6 — t c** cos i (a + ci), (1) 

where the specification is on the Lagrangian plan (Art. 16), viz. a, 6 are two 
parameters serving to identify a particle, and x, y are the co-ordinates of this 
particle at time t. The constant h determines the wave-length, and c is the 
velocity of the waves, which are travelling in the direction of x-negative. 

To verify this solution, and to determine the value of c, we remark, in 
the first place, that 

iti-'-«-. <^) 

* Stokee, {.e. ante p. 400. Another very simple proof of this statement has heen given by 
Rayleigh, ^e. atUe p. 252. 

t Professor of Mathematics at Prague, 1789-1823. His paper, "Theorie der Wellen," was 
published in the Ahh. d. k, bdhm. Oes. d, Wiss. 1802 [Qilbert's AnnaUn d. Physik, t. xzxit (1809)]. 

X "On the Exact Form of Waves near the Surface of Deep Water/' PhU, Trans. 1863 
[Papers, p. 481]. 



I 



,.(3) 



280-261] Oerstner'a Waves 413 

so that the Lagrangian equation of continiiity (Art. 16 (2)) JB satisfied. Again, 
sabstitatijig from (1) in the equations of motion (Art. 13), we find 

a(f + ^)" fe's"*™ *(« + «). 

A (- + w) *"'e" COS i (o + «) + fci'e"" ; 

whence 

^ = const. - 5 Jfc - T e** cos A: (a + cf)[ - c*c" cos k(a+ ct) + Jc»e"». 

W 

For a particle on the free aurface the pieesure mnst be constant; this 
requires 

"■-I c^' 

as in Art. 229. This makes 

^ = cODBt. '-gb + ic»fi»» (6) 

It is obvious from (1) that the path of any particle (a, &) is a circle of 
radios ir>e**. 



The figure shews the forms of the lines of equal pressure b = const., for 
a series of equidistant values of b*. These curves are trochoids, obtained by 

* The diagiam U very simil&r to the one giTen originally bj Qentner, and copied more or 
ess closely by mbsequent writen. A version of Qeratner's investigation, inoluding in imi» 
respect a correction, wM given in the seoond edition of thi« work. Art. 233. 



414 Surface Waves [chap, ix 

rolling circles of radii Ic^ on the under sides of the lines y = 6 4- A~"^, the 
distances of the tracing points from the respective centres being Ar^e**. Any 
one of these lines may be taken as representing the free surface, the extreme 
admissible form being that of the cycloid. The dotted lines represent the 
successive forms taken by a line of particles which is vertical when it passes 
through a crest or a trough. 

It has already been stated that the motion of the fluid in these waves is 
rotational. To prove this, we remark that 

= ISfe** sin i (a + c<)} -f- ce«* Sa, ' (7) 

which is not an exact differential. 

The circulation in the boundary of the parallelogram whose vertices 
coincide with the particles 

{a,yh\ (a + 8a, 6), (a, 6 -f- 86), {a + 8o, 6 4- 86) 

is, by (7), - ^ (ce^** 8a) 86, 

and the area of the circuit is 

Ij^ 8a86 = (1 - e"*) 8a86. 
3 (a, 6) 

Hence the vorticity (co) of the element (a, 6) is 

2kce^ 
^ = -i3^*b (8) 

This is greatest at the surface, and diminishes rapidly with increasing depth. 
Its sense is opposite to that of the revolution of the particles in their circular 
orbits. 

A system of waves of the present type cannot therefore be originated 
from rest, or destroyed, by the action of forces of the kind contemplated in 
the general theorem of Arts. 17, 33. We may however suppose that by 
properly adjusted pressures applied to the surface of the waves the liquid is 
gradually reduced to a state of flow in horizontal lines, in which the velocity 
(m') is a function of the ordinate (y') only*. In this state we shall have 
dx'jda == 1, while y' is a function of 6 determined by the condition 

9 («', y') d (x, y) 



d (a, 6) a (a, 6) ' 



(9) 



or g6 ='~^*** ^^^^ 

* For a fuller statement of the aigument see Stokes' PaperSj t. i. p. 222. 



251-252] Gerstner's Waves 415 

■^"■""^ ■%-%%--^%-^'-"- <"> 

and therefore u' = ce^w (12) 

Hence, for the genesis of the waves by ordinary forces, we require as a 

foundation an initial horizontal motion, in the direction opposite to that of 

propagation of the waves ultimately set up, which diminishes rapidly from 

the surface downwards, according to the law (12), where 6 is a function of y' 

determined by 

y' = 6 - Ik-^e^^ (13) 

It is to be noted that these rotational waves, when established, have zero 
momentum. 

252. Scott Russell, in his interesting experimental investigations*, was 
led to pay great attention to a particular type which he called the 'solitary 
wave.' This is a wave consisting of a single elevation, of height not necessarily 
small compared with the depth of the fluid, which, if properly started, may 
travel for a considerable distance along a uniform canal, with little or no 
change of type. Waves of depression, of similar relative amplitude, were 
found not to possess the same character of permanence, but to break up into 
series of shorter waves. 

Russell's * solitary ' type may be regarded as an extreme case of Stokes' 
oscillatory waves of permanent type, the wave-length being great compared 
with the depth of the canal, so that the widely separated elevations are 
practically independent of one another. The methods of approximation 
employed by Stokes become, however, unsuitable when the wave-length 
much exceeds the depth; and subsequent investigations of solitary waves 
of permanent type have proceeded on different lines. 

The first of these was given independently by Boussinesq| and Rayleigh J. 
The latter writer, treating the problem as one of steady motion, starts 
virtually from the formula 

<!>']- uls = F(X'hiy)=^ e^^ F (x), (1) 

where F {x) is real. This is especially appropriate to cases, such as the 
present, where one of the family of stream-lines is straight. We derive 
from (1) 

<^ = l'-|-jl'" + |^Fv-..., = yl?''-|jF"-f.|^J^-...,..(2) 

where the accents denote differentiations with respect to x. The stream-line 
= here forms the bed of the canal, whilst at the free surface we have 

♦ "Report on Waves," BriL Ass. Rep. 1844. 
t Comptes RenduSf June 19, 1871. 
t Ic. ante p. 252. 



416 ^rface Waves [chap, ix 

^ = — ch, where c is the uniform velocity, and h the depth, in the parts of 
the fluid at a distance from the wave, whether in front or behind. 

The condition of uniform pressure along the free surface gives 

w* + V* = c« - 2sr (y - A), (3) 

or, substituting from (2), 

J'2 _ y2j/j''' ^ y2j''2 + _ . = c« - 2<7 (y - A) (4) 

But, from (2) we have, along the same surface, 

y^'-|^^'"+ ... = -cA (5) 

It remains to eliminate F between (4) and (5) ; the result will be a differential 
equation to determine the ordinate y of the free surface. If (as we will 
suppose) the function F' (x) and its differential coefficients vary so slowly 
with X that they change only by a small fraction of their values when x 
increases by an amount comparable with the depth A, the terms in (4) and 
(5) will be of gradually diminishing magnitude, and the elimination in 
question can be carried out by a process of successive approximation. 

Thus, from (5) 

and if we retain only terms up to the order last written, the equation (4) 
becomes 

■ 

or, on reduction, 

1 2y" ly'*_ 1 igjy-h) 

y* S y 3 y» h* c*h* ^ ' 

If we multiply by y", and integrate, determining the arbitrary constant so 
as to make y' = ioi y = h, we obtain 

1 ly'» 1 y-h gjy-hy 
y S y A "^ A» c»A» ' 



or 



/• = 3^X^-f) («) 



Hence y' vanishes only for y = A and y = c^/g, and since the last factor 
must be positive, it appears that c*/g is a maximum value of y. Hence the 
wave is necessarily one of elevation only, and, denoting by a the maximum 
height above the undisturbed level, we have 

c^^g{h + a) (9) 

which is exactly the empirical formula for the wave-velocity adopted by 
Russell. 



252] 



The Solitary Wave 



417 



The extreme form of the wave must, as in Art. 250, have a sharp crest of 
120° ; and since the fluid is there at rest we shall have c* = 2ga. If the 
formula (9) were applicable to such an extreme case, it would follow that 
a = A. 



If we put, for shortness, 






(10) 



we find, from (8), 



X 



(11) 



(12) 



v=±i(i-5)*. 

the integral of which is 

71 = a sech* \ ^ 

if the origin of x be taken beneath the summit. 

There is no definite 'length ' of the wave, but we may note, as a rough 
indication of its extent, that the elevation has one-tenth of its maximum 
value when xjh = 3'636. 




The annexed drawing of the curve 

y = 1 + ^sech*|a; 

represents the wave-profile in the case a = \Ji. For lower waves the scale 
of y must be contracted, and that of x enlarged, as indicated by the annexed 
table giving the ratio 6/A, which determines the horizontal scale, for various 
values of ajh. 

It will be found, on reviewing the above investigation, that 
the approximations consist in neglecting the fourth power of 
the ratio (A + a)/26. 

If we impress on the fluid a velocity — c parallel to x we 
get the case of a progressive wave on still water. It is not 
difficult to shew that, if the ratio ajh be small, the path of 
each particle is then an arc of a parabola having its axis 
vertical and apex upwards*. 

It might appear, at first sight, that the above theory is 
inconsistent with the results of Art. 187, where it was argued that a wave of 

* Boussinesq, /.c. 



ajh 
•1 


hjh 


1-916 


•2 ' 1-414 


•3 1 1-202 


•4 


1-080 


•6 


1000 


•6 


-943 


•7 


•900 


•8 


.866 


-9 


-839 


1-0 


•816 



L. H. 



27 



418 Surface Waves [chap, ix 

finite height whose length is great compared with the depth must inevitably 
suffer a continual change of form as it advances, the changes being the more 
rapid the greater the elevation above the undisturbed level. The in- 
vestigation referred to postulates, however, a length so great that the vertical 
acceleration may be neglected, with the result that the horizontal velocity 
is sensibly uniform from top to bottom (Art. 169). The numerical table 
above given shews, on the other hand, that the longer the 'solitary wave ' is, 
the lower it is. In other words, the more nearly it approaches to the 
character of a * long ' wave, in the sense of Art. 169, the more easily is the 
change of type averted by a slight adjustment of the particle- velocities*. 

The motion at the outskirts of the solitary wave can be represented by a very simple 
formula. Considering a progressive wave travelling in the direction of x- positive, and 
taking the origin in the bottom of the canal, at a point in the front part of the wave, we 
assume 

<t> =^e-'»(*-«') cos my (13) 

fhis satisfies v'<^ =0.. and the surface-condition 



|t^^|=« (^*) 



will also be satisfied for ?/ = ^, provided 

, -tanm^ ^ . 

'"-^^-mh- <^'^' 

This will be found to agree approximately with Rayleigh's investigation if we put m =6~^ 

The above remark, which was kindly communicated to the author by the late Sir George 
Stokesf , was suggested by an investigation by McCowanJ , who shewed that the formula 

? — ^= -(x+iy) -fa tanh J m(a;-fty) (16) 

c 

satisfies the conditions very approximately, provided 

c« = ^ tan mA, (17) 

m 

and wki = f sin«m(A + |o), a =a tan Jm (A -fo), (18) 

where a denotes the maximum elevation above the mean level, and a is a subsidiary 
constant. In a subsequent paper § the extreme form of the wave when the crest has a 
sharp angle of 120° was examined. The limiting value of the ratio ajh was found to be -78, 
in which case the wave- velocity is given by c* = l-56flrA. 

253. By a slight modification the investigation of Rayleigh and 
Boussinesq can be made to give the theory of a system of oscillatory waves 
of finite height in a canal of limited depth 



* Stokes. "On the Highest Wave of Uniform Propagation," Proc. Camb. Phil. Soc, t. iv. 
p. 361 (1883) [Papera, t. v. p. 140]. 

t Cf. Papers, t. v. p. 162. 

t "On the Solitary Wave," Phil. Mag. (5), t. xxxii. p. 46 (1891). 

§ "On the Highest Wave of Permanent Tjrpe," PhU. Mag. (5), t. xxxviii. p. 361 (1894). 

II Korteweg and De Vries, "On the Change of Form of Long Waves advancing in a Rect- 
angular Canal, and on a New Type of Long Stationary Waves," Phil. Mag. (6), t. zxxix. p. 422 
(1896). The method adopted by these writers is somewhat different. Moreover, as the title 



252-253] Theory of Korteweg and De Vries 419 

In the steady-motion form of the problem the momentum per wave-length (X) is repre- 
sented by 

j jpudxdy= -p j j ^^^y= -P^i^* (1^) 

where ^^ corresponds to the free surface. If A be the mean depth, this momentum may be 
equated to pch\, where c denotes (in a sense) the mean velocity of the stream. On this 
understanding we have, at the surface, V^i = - ch, as before. The arbitrary constant in 
(3), on the other hand, must be left for the moment undetermined, so that we write 

u^+v^=C-2gy, (20) 

We then find, in place of (8), 

tr=^t{y-l){hi-y){y-h,) (21) 

where k^ , ^2 ^^^ ^^^ upper and lower limits of y, and 

'=.^ : ^''^ 

It is implied that I cannot be greater than A,. 

If we now write y=^ cos* x + ^ s^* X> (2^) 

we find ^^ = V{1 -ifc*sin«x}, (24) 

'^Vl^*;^}' *'=*t^ ^^> 

Hence, if the origin of x be taken at a crest, we have 

and y=K +(^1 -^2) en* -. [mod. k] (27)* 



The wave-length is given by 

V(l-ik*sin*x) 



^=2^^" ^Ti-fcaTx =2^^i W (28) 



Again, from (23) and (24), 

j\d.=2fffJ'^-^';'l^^.^^dx=mI'^{k)Hh^-l)EAk)} (29) 

Since this must be equal to AX, we have 

(h-l) Fi [k) =(^ -0 El (Jfc) (30) 

In equations (25), (28), (30) we have four relations connecting the six quantities ^, ^, 
/, k, X, fif so that if two of these be assigned the rest are analytically determinate. The 
wave- velocity c is then given by (22) f- For example, the form of the waves, and their 
velocity, are determined by the length X, and the height A^ of the crests above the bottom. 

The solitary wave of Art. 252 is included as a particular case. If we put l=h^, we 
have k = l, and the formulae (28) and (30) then shew that X = x , A, = A. 

indicates, the paper includes an examination of the manner in which the wave-profile is changing 
at any instant, if the conditions for permanency of type are not satisfied. 

For other 'modifications of Rayleigh*8 method reference may be made to Gwyther, Phil, 
Mag, (6), t. 1. pp. 213, 308, 349 (1900). 

* The waves represented by (27) are called *cnoidal waves* by the authors cited. For the 
method of proceeding to a higher approximation we must refer to the original paper. 

t When the depth is finite, a question arises as to what is meant exactly by the * velocity of 
propagation.' The velocity adopted in the text is that of the wave-profile relative to the centre 
of inertia of the mass of fluid included between two vertical planes at a distance apart equal to 
the wave-length. Cf. Stokes, Papers, t. i. p. 202. 

27—2 



420 Surface Waves [chap, ix 

254. The theory of waves of permanent type has been brought into 
relation with general dynamical principles by Helmholtz*. 

If in the equations of motion of a 'gjrrostatic ' system, Art. 141 (24), we 
put 

where V is the potential energy, it appears that the conditions for steady 
motion, with q^, q^, ... g„ constant, are 

l(F + iC) = 0, ^(7 + X) = o, ..., ^(F + iC) = o,..(2) 
dqi dq2 oq^ 

where K is the energy of the motion corresponding to any given values of 
the co-ordinates q^, jj* • • • ?n when these are prevented from varying by the 
application of suitable extraneous forces. 

This energy is here supposed expressed in terms of the constant momenta 
corresponding to the ignored co-ordinates x* x'» • • • > *^^ ^^ ^^^ palpable 
co-ordinates Ji, y2> • • • ?n« I* ^^7 however also be expressed in terms of the 
velocities x> x'» • • • *^^ *^® co-ordinates 9], 92> • • • 9n > i^ ^^ form we denote 
it by Tq. It may be shewn, exactly as in Art. 142, that dTo/dqr = — dK/dq^ 
so that the conditions (2) are equivalent to 

|-(7-2'o) = 0, i^^(y-To) = 3^-^(7- To) = 0. ..(3) 

Hence the condition for free steady motion with any assigned constant 
values of ji, jj, ... j„ is that the corresponding value of 7 + JT, or of 7 — Tq* 
should be stationary. Cf. Art. 203 (11). 

Further, if in the equations of Art. 141 we write — dV/dqr + Qr for Q^y so 
that Qr now denotes a component of extraneous force, we find, on multiplying 
by ji, 92> • • • 9n i^ order, and adding, 

where "ST is the part of the energy which involves the velocities ^i, ^2> • • • 9n- 
It follows, by the same argument as in Art. 205, that the condition for 
* secular ' stability, when there are dissipative forces affecting the co-ordinates 

?i> ?2> • • • ?n> but not the ignored co-ordinates x, x'> • • • > i® *^^ V + K should 
be a minimum. 

In the application to the problem of stationary waves, it will tend to 
clearness if we eliminate all infinities from the question by imagining that 
the fluid circulates in a ring-shaped canal of uniform rectangular section (the 

* "Die Energie der Wogen und des Windes," Berl Monaisber, July 17, 1890 [Wisa, Abh. 
t. iii. p. 333]. 



254] Dynamical Condition for Permanent Type 421 

sides being horizontal and vertical), of very large radius. The generalized 
velocity x corresponding to the ignored co-ordinate may be taken to be the 
flux per unit breadth of the channel, and the constant momentum of the 
circulation may be replaced by the cyclic constant k. The co-ordinates 
?i> ?2> • • • ?n of the general theory are now represented by the value of the 
surface-elevation (tjI) considered as a function of the longitudinal space- 
co-ordinate X, The corresponding components of extraneous force are repre- 
sented by arbitrary pressures applied to the surface. 

If I denote the whole length of the circuit, then considering unit breadth 
of the canal we have 



F 

' 

where r^ is subject to the condition 



= \gp\ -qHx, (5) 

J 



/ y^dx = 0. 



(6) 



If we could with the same ease obtain a general expression for the kinetic 
energy of the steady motion corresponding to any prescribed form of the 
surface, the condition in either of the forms above given would, by the usual 
processes of the Calculus of Variations, lead to a determination of the possible 
forms, if any, of stationary waves*. 

Practically, this is not feasible, except by methods of successive approxi- 
mation, but we may illustrate the question by reproducing, on the basis of 
the present theory, the results already obtained for 'long ' waves of infinitely 
small amplitude. 

If h be the depth of the canal, the velocity in any section when the surface is maintained 
at rest, with arbitrary elevation 17, is x/(* +'?)» where x is the flux. Hence, for the cyclic 
constant, 

«=x/V+,)-><fe=^|(l4jV&:) (7) 

approximately, where the term of the first order in 17 has been omitted, in virtue of (6). 

The kinetic energy, \pKXi may be expressed in terms of either x or #c. We thus obtain 
the forms 






* For some general considerations bearing on the problem of stationary waves on the common 
surface of two currents reference may be made to Helmkoltz' paper. This also contains, at 
the end, some speculations, based on calculations of energy and momentum, as to the length of 
the waves which would be excited in the first instance by a wind of given velocity. These appear 
to involve the assumption that the waves will necessarily be of permanent type, since it is only 
on some such hypothesis that we get a determinate value for the momentum of a train of waves 
of small amplitude. 



422 Surface Waves [chap, ix 

The variable part of V -Tq is 



and that of F + jST is 



^'{'-w)jy^ ^''^ 



It is obvious that these are both stationary for 17 =0; and that they will be stationary 
for any infinitely small values of »;, provided x*=gh^, or K^=ghl^, K we put x=^^» ^^ 
jc =c/, this condition gives 

c^-=gh, (12) 

in agreement with Art. 175. 

It appears, moreover, that 1; =0 makes V +K a maximum or a minimum eu^cording as 
c' is greater or less than gh. In other words, the plane form of the surface is secularly 
stable if, and only if, c<J{gh). It is to be remarked, however, that the dissipative forces 
here contemplated are of a special character, viz. they affect the vertical motion of the 
surface, but not (directly) the flow of the liquid. It is otherwise evident from Art. 176 
that if pressures be applied to maintain any given constant form of the surface, then if 
c^>gh these pressures must be greatest over the elevations and least over the depres- 
sions. Hence if the pressures be removed, the inequalities of the surface will tend to 
increase. 

Wave-Propagation in Two Dimensions. 

255. We may next consider some cases of wave-propagation in two 
horizontal dimensions x, y. The axis of z being drawn vertically upwards, 
we have, on the hypothesis of infinitely small motions, 

2 = |-,. + ^(0. (1) 

where <f> satisfies V ^ = (2) 

The arbitrary function F {t) may be supposed merged in the value of d^jdt. 

If the origin be taken in the undisturbed surface, and if J denote the 
elevation at time t above this level, the pressure-condition to be satisfied at 
the surface is 

'-j[tL «'> 

and the kinematical surface-condition is 



dt 
cf. Art. 227. Hence, for z = 0, we must have 



'^-[lU <« 



8/« ' ^ dz 
or, in the case of simple-hannonic motion, 



S + ?^ = 0. (5) 



<rV = i7|t' (^) 



if the time-factor be e» (»'+'). 



254-255] Wave-Propagatio7i in Two Dimeiisiojis 



423 



The fluid being supposed to extend to infinity, horizontally and down- 
wards, we may briefly examine, in the first place, the effect of a local initial 
disturbance of the surface, in the case of symmetry about the origin. 

The typical solution for the case of initial rest is easily seen, on reference 
to Art. 100, to be 



provided 

as in Art. 228. 



. sinai fc« T /7 \ 

J = cos at Jq (kw), J 
^^ = 9h 



(7) 



(8) 



To generalize this, subject to the condition of symmetry, we have 
recourse to the theorem 



r 00 /•« 

JO ^0 



(9) 



JO 

of Art. 100 (12). Thus, corresponding to the initial conditions, 

i =/(«>). '^0 = 0, (10) 



/•GO 'a /• OO 

we have ^z=g\ 6** Jq (feo) icZA; | f{a)jQ{ka)ada,\ . 

J <y ^0 I 

J = I COS at Jq {Jew) kdk I f (a) Jq (ka) ada, 

Jo Jo ) 



....(11) 



If the initial elevation be concentrated in the immediate neighbourhood 
of the origin, then, assuming 

f"V(a) ^ada =1, (12) 

J 

we have <f> = 2^/* ^^^ «** «^o (*tD) kdk (13) 

Expanding, and making use of (8), we get 

If we put z = —r cos 0, w = rsind, (15) 

/.« -^ 
we have I e** Jq (kw) dk = -, (16) 

by Art. 102 (9), and thence* 

j\^jQikw)kr^dk=[l-Jl==n\^^ (17) 



* HobBon, Proc. Lond. Math, Soc. t. xxv. pp. 72, 73. This formula may, however, be 
dinpensed with; see the first footnote on p. 374 ante. 



424 Surface Waves [chap, ix 

where ^ = cos 6 (cf. Art. 85). Hence 

From this, the value of J is to be obtained by (3). It appears from 
Arts. 84, 85 that 

^2n+i(0) = 0, P,n(0) = (-)- ^'2'4".!^.''2n^^ > ••••(^^^ 



whence 



^~2«i»M2!nj 6! U/ 10! UJ •••[•'• 



(20) 



« 



It follows that any particular phase of the motion is associated with 
a particular value of gi^lm, and thence that the various phases travel radially 
outwards from the origin, each with a constant acceleration. 

No exact equivalent for (20), analogous to the formula (21) of Art. 238 
which was obtained in the two-dimensional form of the problem, and accord 
ingly suitable for discussion in the case where gt^jw is large, has been dis- 
covered. An approximate value may however be obtained by Kelvin's method 
(Art. 241). Since J^ {z) is a fluctuating function which tends as z increases 
to have the same period 27r as sin z^ the elements of the integral in (13) will 
for the most part cancel one another with the exception of those for which 

t da/dk = ro, or Aro == gt^im (21) 

nearly. Now when Jew is large we have 

•^0 (^) = ( J^)* sin (kw + iTT), (22) 

approximately, by Art. 194 (15), and we may therefore replace (13) by 

<f> = -i-^, -T [ e^ cos {<yt - kw - in) dk (23) 

Comparing with (6) and (8) of Art. 241, and putting now 2 = 0, we find as 
the surface value of <f> 



6o = i — ^ sin (at — Aro), (24) 

27rw^V \td^<^ldk^\ 

where k and a are to be expressed in terms of w and t by means of (8) and 
(21). Note has here been taken of the fact that d^/dk^ is negative. Since 

at = (gkt^)^ = 2kw, t d^ajdk^ = - Ighk'^ = - 2ro«/^«, . . (25) 
we have <*« = -~ — sin ?^ - (26) 

♦ This result was given by Cauchy and Poiason. 



255] Effect of a Local Impulse 425 

The surface elevation is then given by (3). Keeping, for consistency, only 
the most important term, we find 

«= -/^cosg, (27) 

which agrees with the result obtained, in other ways, by Cauchy and Poisson. 

It is not necessi^ry to dwell on the interpretation, which will be readily 
understood from what has been said in Art. 240 with respect to the two- 
dimensional case. The consequences were worked out in some detail by 
Poisson on the hypothesis of an initial paraboloidal depression. 

When the initial data are of impulse, the typical solution is 

p^ = cos aie^ Jq (Arm), 



1 = sin (7^ Jq (km), 

9P 



(28) 



which, being generalized, gives, for the initial conditions 

P<l>o='F(m), J = (29) 

the solution 

<f> = - I COS aie^ Jq {Jew) kdk j F (a) Jq {ka) ada, 

^•'^ -'^ I ....(30) 

1 TOO /-OO ' ^ ' 

^ = 1 a sin o/ Jo (^) *^* 1 ^ (o) Jo (^) ada. 

In particular, for a concentrated impulse at the origin, such that 

f F (a) 27rada =^ 1, (31) 

J 

1 f* 

we find <f> = 5 — I cos <rfe*» Jq (kw) kdk (32) 

ZnrpJ 

Since this may be written 

^ = 2^I/I^«"'^»(*«')*'^^' •• (33) 

we find, performing 1/gp . d/dt on the results contained in (18) and (20), 

^ 1 If, (ft) gt* 2\P, (ft) (g<')« 3 ! f 3 (/*) ) \ 

^ 27rpl r* 2! f» "^ 4! r^ •••|'| 



2iT/)tD» (.* 5! Vro/ ' 9! (toj •"}• ) 
Again, when ^gt'/m is large, we have, in place of (27), 

2*w/)m* 



C=-::/^sing (35) 



42« Surface Waves [chap, ix 

256. We proceed to consider the effect of a local disturbance of pressure 
advancing with constant velocity over the surface. This will give us, at all 
events as to the main features, an explanation of the peculiar system of waves 
which is seen to accompany a ship moving through sufl&ciently deep water. 

A complete investigation, after the manner of Arts. 242, 243, would be 
somewhat difficult ; but the general characteristics can readily be made out 
with the help of preceding results, the procedure being similar to that of 
Art. 249. 

Let us suppose that we have a pressure-point moving with velocity c 
along the axis of x, in the negative direction, and that at the instant under 
consideration it has reached the point 0. The elevation J at any point P may 
be regarded as due to a series of infinitely small impulses applied at equal 
infinitely short intervals at points of the axis of x to the right of 0. Of the 
annular wave-systems thus successively generated, those only will combine 
to produce a sensible effect at P which had their origin in the neighbourhood 
of certain points (?, which are determined by the consideration that the phase 
at P is 'stationary ' for variations in the position of Q, Now if t is the time 
which the source of disturbance has taken to travel from Q to 0, the phase 
of the waves at P, originated at Q, is 

S + i'^' (1) 

where w = QP (Art. 255 (35)). Hence the condition for stationary phase is 

^-Y <2) 




Since, in this differentiation, and P are regarded as fixed, we have 

isr = c cos By 
where 6 = OQP ; hence 

Og= c^ = 2© sec fl (3) 

It is further evident that the points in the immediate neighbourhood 
of P, for which the resultant phase is the same as at P, will lie in a line 
perpendicular to QP. A glance at the above figure then shews that a curve 



256] 



Waves due to a Travelling Disturbance 



427 



of uniform phase is characterized by the property that the tangent bisects 
the interval between the origin and the foot of the normal. If p denote 
the perpendicular from the origin to the tangent, and 6 the angle which p 
makes with the axis of x, we have, by a known formula, 



whence 



2y=-gcot5, 
y = a cos^ 0, 



(5) 




The forms of the curves defined by (5) are shewn in the annexed figure*, 
which is traced from the equations 



x = y cos fl — -j^^ sin = Ja (5 cos — cos 3&), 
y = psinO -\- -j^ cos fl = — Ja (sin -\- sin 30), 



(6) 



The phase-difference from one curve to the corresponding portion of the next 
is 2n, This implies a difference 27rc^lg in the parameter a. 

Since two curves of the above kind pass through any assigned point P 
within the boundaries of the wave-system, it is evident that there are tioo 
corresponding effective positions of Q in the foregoing discussion. These are 
determined by a very simple construction. If the line OP be bisected in C, 

* Cf. Sir W. Thomson, "On Ship Waves," Proe. Irust Mech. Eng. Aug. 3, 1887 [Popular 
Lectures, t. iii. p. 482], where a similar drawing is given. The investigation there referred to, 
based apparently on the theory of * group- velocity/ was not published. See also R. E. Froude 
"On Ship Resistance," Papers of the Greenock Phil Soc. Jan. 19, 1894. 



428 



Surface Waves 



[chap. IX 



and a circle be drawn on CP as diameter, meeting the axis of xmR-i^R^, 
the perpendiculars PQi, PQ2 to PJBi, PR2, respectively, will meet the axis 
in the required points Q^, Q^. For CRi is parallel to PQi and equal to 
i-PQi > tl^c perpendicular from on PRi produced is therefore equal to PQi . 
Similarly, the perpendicular from on PR2 produced is equal to PQ^, 




The points Q^ , Q^ coincide when OP makes an angle sin-^ J, or 19° 28', with 
the axis of symmetry. For greater inclinations of OP they are imaginary. 
It appears also from (6) that the values of re, y are stationary when sin' = ^ ; 
this gives a series of cusps lying on the straight lines 

1 



X 



2V2 



= ± tan 19° 28'. 



(7) 



It will be seen immediately that a change of phase takes place from one 
portion of a curve to the other at the cusps. 

To obtain an approximate estimate of the actual height of the waves, 
in the different parts of the system, we have recourse to t6e formula (35) of 
Art. 255. If Pq denote the total disturbing pressure, the elevation at P due 
to the annular wave-system started at a point Q to the right of may be 
written 



«S = - 8 V2.p^* • «- £ • ^o«'' 



(8) 



where 



tn = pg, 



t = 



OQ 



This is to be integrated with respect to t, but (as already explained) the only 
parts of the integral which contribute appreciably to the final result will be 
those for which t has very nearly the values (tj, tj) corresponding to the 
special points Q^, Q2 &bove mentioned. 

As regards the phase, we have, writing t = r + t\ 



9^ 

4m 



'[tMiov&Aum^ <»' 

where, in the terms in [ ], i is to be put equal to r^ or tj as the case may be. 



256] Ship- Waves 429 

The secoad term vanishes by hypothesis, since the phase at P for waves 
started near Q^ or Q, ^ * stationary.' Again, we find 

Since «• = c cos d, w = , (10) 

w 

this gives, with the help of (2), 



[s>(£)]=l<»--°-« '"' 



Owing to the fluctuations of the trigonometrical term no great error 
will be committed if we neglect the variation of the first factor in (8), or if, 
further, we take the limits of integration with respect to t' to be ± oo . 
We have then, approximately, 

j-8^./:.-'°(s:+».^")*- 

where V = g^ (i - tan« dj), m,« = g^- (tan« », - i), (13) 

and the suffixes refer to the points Qi, Q, of the last figure. 

Since f cos mH'*dt' = [ sin mH'*dt' = V(if^)lm, (14) 

where the positive value of m is understood, we find 



i= 



(15) 

The two terms give the parts due to the transverse and lateral waves 
respectively. Since tOi = PQi = ^ct^ cos &i , tn, = PQ2 ==^0x2 cos flg? it appears 
that if we consider either term by itself, the phase is constant along the corre- 
sponding part of the curve 

p = xo = a cos* By 
whilst the elevation varies as 

\/2g*fo sec»g 

^Vc»a*V|l-38in*e| ^'^^ 



430 Surface Waves [chap, ix 

At the cusps, where the two systems combine, there is a phase-difference 
of a quarter- period between them*. 

The formulae make f infinite at a cusp, where sin* 5 = ^, but this is 
merely an indication of the failure of our approximation. That the elevation 
at a point P in the neighbourhood of a cusp would be relatively great might 
have been foreseen, since, as appears from (9) and (11), the range of points 
on the axis of x which have sent waves to P in sensibly the same phase is 
then abnormally extended. 

The infinity which occurs when 6 = \'tt is of a somewhat different 
character, being due to the artificial nature of the assumption we have made, 
of a pressure concentrated at a jxyint. With a diffused pressure this difficulty 
would disappear. 

It is to be noticed, moreover, that the whole of this investigation applies 
only to points for which gt^jAm is large; cf. Arts. 240, 255. It will be 
found on examination that this restriction is equivalent to an assumption 
that the parameter a is large compared with 27rc^/g. The argument there- 
fore does not apply without reserve to the parts of the wave-pattern near the 
origin. 

Although the mode of disturbance is different, the action of the bows of 
a ship may be roughly compared to that of a pressure-point of the kind we 
have been considering. The figure on p. 427 accounts clearly for the two 
systems of transverse and lateral waves which are in fact observed t, and 
for the especially conspicuous * echelon ' waves at the cusps, where these two 
systems coalesce. These are well shewn in the annexed drawing J by 
Mr R. E. Froude of the waves produced by a model. 

A similar system of waves is generated at the stern of the ship, which 
may roughly be regarded as a negative pressure-point. With varying speeds 
of the ship the stern-waves may tend partially to annul, or to reinforce, 

* The investigation in the preceding edition was defective through an oversight, but was 
corrected in the German translation, Leipzig, 1907. Kelvin returned to the subject in 1905: 
"Deep Sea Ship- Waves," Trans. E. 8. Edin. t. xxv. p. 1060 [Papers, t. iv. p. 407]. The distri- 
bution of surface elevation ai rived at differs however from that found in the text, owing to the 
adoption of a special law of pressure-intensity which is unfortunately not an adequate repre- 
sentation of a localized disturbance. See also Havelock, "The Propagation of Groups of Waves 
in Dispersive Media. . .," Proc. Roy, Soc. A, t. Ixzxi. p. 398 (1908). The waves due to a local 
obstacle at the bottom of a stream were investigated by Ekman, "On Stationary Waves in 
Running Water," Arkiv for Matem, t. iii. (1906). 

f A diagram of the forms of the wave-ridges, in which account is taken of the change of 
phase at the cusps, is given by Ekman, l.c. The whole system is best seen when viewed almost 
vertically from a great height on the precipitous .sides of a lake (such as Garda). If the water 
be quite smooth except for the passage of a steamer, the wave-pattern is seen beautifully 
developed in accordance with the diagram. 

X Copied, by permission of Mr Froude and the Council of the Institute of Naval Architects, 
from a paper by the late W. Froude, "On the Effect on the Wave-Making Resistance of Ships of 
Length of Parallel Middle Body," Trans. Inst. Nav. Arch. t. xvii. (1877). 



256] Ship- Waves 431 

the effect of the bow-waves, and consequently the wave-resbtance to the 
ship as a whole for a given speed may fluctuate up and down as the length 
of the ship is increased*. Cf. Art. 24i. 



To examine the modification produced in the wave-pattern when the 
depth of the water has to be taketi into account, the preceding argiunent 
must be put in a more general form. If, as before, ( is the time the 
pressure-point has taken to travel from Q to 0, it may be shewn that the 
phase of the disturbance at P, due to the impulse delivered at Q, will differ 
only by a constant from 

k{Vt-m), (17) 

if 'i/njk be the predominant wave-length in the neighbourhood of P, and 
F the corresponding wa ve- velocity f. This predominant wave-length is 
determined by the condition that the phase is stationary for variations of 
the wave-length only, i.e. 

^.*(F(-nj) = 0, or x?=Vl, (18) 

where V, = d (kVydk, is the group- velocity (Art. 236). 

For the effective part of the disturbance at P, the phase (17) must 
further be stationary as regards variations in the position of Q; hence, 
differentiating partially with respect to (, we have 

w=V, or V = cco69 (19) 

• See W. Froude, I.c. and R, E. Proude, "On the Leading Phenomena of the Wave-Making 

Resistance of Ships," Traru. Intl. Nav. Arch. t. iiii. (1S61), where drawings of actual wave- 
patterns under varied conditions of speed are given, which are, as to their main features, in 
atriliing agreement witb the results of the above theory. Some of these drawings are reproduced 
in Kelvin's paper in the Proc. Irul. Mteh. Eng. above cited. 

For a discussion of the wave- resistance encountered by an ideal form of ship see Michell, 
Fha. Mag. (6), t. ilv. p. 106 (1898). 

t The symbol e,, which was previously employed in this seiwe, now denotes the velocity of 
tbe pressure-point over the water. 



432 Surface Waves [chap, ix 

since m^ c cos 0. Now, referring to the figure on p. 426, we have 

p = ce cos fl - to = F< - n; (20) 

Hence for a given wave-ridge p will bear a constant ratio to the wave- 
length A, and in passing from one wave-ridge to the next this ratio will 
increase (or decrease) by unity. Sinq^ A is determined as a function of 
by (19), this gives the relation between p and 0. 

Thus in the case of infinite depth, the formula (19) gives 

c«cosa0= F« = ^, (21) 

and the required relation is of the form 

p = a cos« 0, (22) 

as above. 

When the depth (A) is finite, we have 



and the relation is 



c^cos^fl = F« = f^ tanh ^, (23) 

?tanh - = 4 cos^e, (24) 

a p gfi 



where the values of a for successive wave-ridges are in arithmetic pro- 
gression. The forms of the curves in various cases have been sketched by 
Ekman*. Since the expression on the left-hand side cannot exceed unity, 
it appears that if c* > gh there will be an inferior limit to the value of d, 
determined by 

cos«fl = ^ (25) 

It follows that when the speed of the disturbing influence exceeds '\/{gh) 
the transverse waves disappear, and we have only the lateral waves. 
This tends to diminish the wave-making resistance (cf. Art. 249) f. 



• 



Standing Waves in Limited Masses of Water. 

257. The problem of free oscillations in two horizontal dimensions (x, y), 
in the case where the depth is uniform and the fluid is bounded laterally by 
vertical walls, can be reduced to the same analytical form as in Art. 190. 

If the origin be taken in the undisturbed surface, and if ^ denote the 
elevation at time t above this level, the conditions to be satisfied at the free 
surface are as in Art. 255 (3), (4). 

* l.c. ante p. 430 

t It is found that the power required to propel a torpedo-boat in relatively shallow water 
increases with the speed up to a certain critical velocity, dependent on the depth, then decreases, 
and finally increases again. See papers by Rasmussen, Trans. InsL Nav. Arch. t. xli. p. 12 
(1899); Rota, ibid. t. xlii. p. 239 (1900); Yanow and Marriner, ibid. t. xlvii. pp. 339, 344 (1906). 



256-257] Waves in Limited Masses of Water 433 

The equation of continuity, V^ == 0, and the condition of zero vertical 
motion at the depth 2; = — A, are both satisfied by 

<f> = <f>i cosh A (2 4- A), (1) 

where <f>i is a function of x, y, such that 

'^' + 'I? + *V. - ..(2) 

The form of <f>i and the admissible values of k are determined by this 
equation, and by the condition that 

, ' ^-0 m 

at the vertical walls. The corresponding values of the * speed' (a) of the 

oscillations are then given by the surface-condition (6), of Art. 255; viz. we 

have 

a* Bs gk tanh kh (4) 

ik 
This makes ? = — sinh kh .<f>i (5) 

The conditions (2) and (3) are of the same form as in the case of small 
depth, and we could therefore at once write down the results for a rectangular 
or a circular* tank. The values of k, and the forms of the free surface, in the 
various fundamental modes, are the same as in Arts. 190, 191 1, but the 
amplitude of the osclQation now diminishes with increasing depth below the 
surface, according to the law (1) ; whilst the speed of any particular mode is 
given by (4). 

When kh is small, we have a* = k^gh, as in the Arts, referred to. 

We may also notice in this connection the case of a ]ong and narrow rectangular tank 
having near its centre one or more cylindrical obstacles, whose generating lines are vertical. 



The origin being taken at the centre of the free surface, and the axis of x parallel to 
the length /, we imagine two planes x = ±z* tohe drawn, such that x' is moderately large 
compared with the horizontal dimensions of the obstacles, whilst still small in comparison 
with the length (/). Beyond these planes we shall have 

S«0, 



car* 



+ k^(t>i =0 (6) 



* For references to the original investigations by Poisson and Rayleigh of waves in a 
circular tank see p. 281. The problem was also treated by Merian, Ucber die Bewegung 
iropfbarer FlUasigkeilen in Offween, Basel, 1828 [see VonderMuhll, Maih. Ann. t. xxvii. 
p. 575] and by Ostrogradsky, "M^moire sur la propagation des ondes dans un bassin 
cylindrique," M4m. des Sav. JStrang, t. iii. (1832). 

t It may be remarked that either of the two modes figured on p. 280 may be easily 
excited by properly-timed horizontal agitation of a tumbler containing water. 

L. H. 28 



434 Surface Waves [chap, ix 

approximately, and therefore, for x>z', 

(Pi^Asiakx+BcoBlcXt (7) 

whilst, for x< -x\ 

<l>i=iABmkx-B cos kx, (8) 

since in the gravest mode, which is alone here considered, 4> must be an odd function of x. 

In the region betiveen the planes x = ±:x' the configuration of the lines 0^= const, is, 
for a reason to be explained in Art. 290 in connection with other questions, sensibly 
the same as if in (2) we were to put k=0. So far as this region is concerned, the 
problem is in fact the same as that of conduction of electricity along a bar of metal which 
has the same form as the actual mass of water, and has accordingly one or more cylindrical 
perforations occupying the place of the obstacles. The electrical resistance between the 
two planes is then equivalent to that of a certain length 2x^ + a of an unperiorated bar of 
the same section. The difference of potential between the phases may be taken to be 
2(kAx' -{-B), by (7), since kx' is small; and the current per unit sectional area is kA, 
approximately. Thus 

2 {kAx' +B)=^(2x'+a)kA, (9) 

whence B/A =ika (10) 

and (t)i=A(Bmkx+ikacoskx), (11) 

for x>x\ 

The condition 80/3a? =0, to be satisfied for x=il, gives 

cos ikl- ika sin H (12) 

or, since ka\s& small quantity, 

cos iifc (^ + a) =0 .(13) 

The introduction of the obstacles has therefore the effect of virtually increasing the 
length of the tank by a. The period of the gravest mode is accordingly 



v=V(7-*-*^^') '''' 



wherer=Z+a. 



The value of a is known for one or two cases. In the case of a circular column of radius 
b, in the centre of the tank, the formulae (11) and (13) of Art. 64 shew that 0i varies as 
x+C, or x+nb^/a, practically, when x is comparable with the breadth a of the tank. 
Comparing with (11) above we see that 

a=2nb*/a, (16) 

subject to the condition that the ratio b/a must not exceed about J*. 

When the plane x=0 is occupied by a thin rigid diaphragm of breadth a, having a 
central vertical slit of breadth c, the formula is 

a=-logsec-^ (16) 

* The formula (14) was in this case found to be in good agreement with experiment (Lamb 
and Cooke, PhU. Mag. (6), t. zx. p. 303 (1910)). The experiments were made chiefly with a view 
to test the above method of approximation, which has other more important applications ; see 
Arts. 306, 307. 



267-258] Waves in Limited Masses of Water 435 

258. The number of cases of motion with a variable depth, of which the 
solution has been obtained, is very small. 

1^. We may notice, first, the two-dimensional oscillations of water across a channel 
whose section consists of two straight lines inclined at 45^ to the vertical*. 

The axes of ^, z being respectively horizontal and vertical, in the plane of a cross- 
section, we assume 

<^+*^=il {cosh A;{y+t2)+oosifc(y+»2)}, (1) 

the time-factor cos (ai+t) being understood. This gives 

<f>=A(QOBhhyQO&kz+<iORhyGo&\ikz)y '^=^4 (sinhJEr^sinJb-sinibysinhib;). ..(2) 

The latter formula shews at once that the lines y = ±.z constitute the stream-line '^=0, 
and may therefore be taken as fixed boundaries. 

The condition to be satisfied at the free surface is, as in Art. 227, 

<^'<^=?^| (3) 

Substituting from (2) we find, if h denote the. height of the siuiace above the origin, 

a^ (cosh ky cos kh + cos ky cosh kh) =gk(- cosh kysinhk-k- cos ky sinh M). 

This will be satisfied for all values of y, provided 

o*' cos kh= -gk sin kh, o-' cosh kh =gk sinh kh, (4) 

whence tanh kh= - tan kh .....' (5) 

This determines the admissible values of k; the corresponding values of o* are then given 
by either of the equations (4). 



Since (2) makes an even function of y, the oscillations which it represents are sym- 
metrical with respect to the medial plane y =0. 



The asymmetrical oscillations are given by 

+ti^ =%A {cosh k {y+iz) - cos k (y +t2)}, (6) 

or <l}= -A{smhkyBmkz+ankyamhkz), ylr=A{coehkycoekz-oo8kyGoshkz). (7) 

The stream-line ifr =0 consists, as before, of the lines y = ^z; and the surface-condition (3) 
gives 

0-' (sinh kyamkh+ sin ky sinh kh) =gk (sinh ky cos kh + sinky cosh kh). 

This requires 

a* sin kh =gk cos kh, a^ sinh kh =gk cosh kh, (8) 

whence tanh kh = tan kh (9) 

The equations (5) and (9) present themselves in the theory of the lateral vibrations of 
a bar free at both ends ; viz. they are both included in the equation 

cos m cosh m = 1, (10)t 

where m=2kh. 

The root kh =0, of (9), which is extraneous in the theory referred to, is now important ; 
it corresponds in fact to the slowest mode of oscillation in the present problem. Putting 

* Kirohhoff, **Ueb6r stehende Sohwingungen einer schweren Flussigkeit,'* Berl. Monatther. 
May 15, 1879 [Ges, Abh. p. 428]; Greenhill, Ic, ante p. 363. 

t Cf. Rayleigh, Theory of Sound, t. i. Art 170, where the numerical solution of the 
equation is fully disoussed. 

28—2 



436 



Surface Waves 



[OHAP. IX 



Al^ -Bj and making k infinitesimal, the formulae (7) become, on lestoring the time-factor, 
and taking the real parts, 

=- 25^8 . cos (rr* + c), i/r=fi{y'-2;*).oos(a'<+€), (11) 



whilst from (8) 

The corresponding form of the free surface is 






(12) 



f=-r^1_ =2o-fi%.sin(c7<+€) (13) 



The surface in this mode is therefore always plane. The annexed figure shews the lines of 
motion (^ = const.) for a series of equidistant values of ifr. 




The next gravest mode is symmetrical, and is given by the lowest finite root of (5), 

which is ^^ = 2*3650, whence o- = 1*6244 {glhy. The profile of the surface has now two nodes, 
whose positions are determined by putting =0, z =^, in (2) ; whence it is found that 

I = ±-5516*. 

The next mode corresponds to the lowest finite root of (9), and so onf. 

2^. Qreenhill, in the paper already cited, has investigated the symmetrical oscillations 
of the water across a channel whose section consists of two straight lines inclined at 60° to 
the vertical. In the (analytically) simplest mode of this kind we have, omitting the time- 
factor, 

+i^ =iA (y+izf +B, (14) 

or <^=ilz(2«-3y«)+fi, Vr=ily (y« -3z«), (16) 

the latter formula making i/r = along the boundary y = ± tJS.z. The surface-condition (3) 
is satisfied for z=h, provided 

<r^=glK B=2Ah^ (16) 

* Rayleigh, Theory of Sound, Art. 178. 

t An experimental verification of the frequencies, and of the positions of the loops (places of 
maximum vertical amplitude), in various fundamental modes, was made by Kirchhoff and 
Hansemann, "Ueber stehende Schwingungen des Wassers,'* Wied, Ann. t. x. (1880) [Kirchhoff, 
Oes, Ahh. p. 442]. 



268-269] Transverse Oscillations in a Canal 437 

The oorrespondin^ form of the free suifaoe, viz. 

f=J[tl-r-'^<**-'^)^'"<''^') ("> 

ia a parabolic cylinder, with two nodes at distances of '6774 of the half-breadth from the 
centre. The slowest mode, which must evidently be of asymmetrical type, has not yet 
been determined. 

3°. If in any of the above cases we transfer the origin to either edge of the canal, and 
then make the breadth infinite, we get a system of standing waves on a sea bounded by a 
sloping bank. This may be regarded as made up of an incident and a reflected system. 
The reflection is complete, but there is in general a change of phase. 

When the inclination of the bank is 46° the solution is 

0=JJ{e*»(oosifcy-ainA:y)+e~*''(cosAa+3inib)} C03{«r<+f) (18) 

For an inclination of 30° to the horizontal we have 

= JI {«*• sin ifcy + c-** <^'>'+*' sin \h (y - V3z) 

- V3e-**<^'>'-*> cos \k (y +V32)} cos (at +0 (19) 

In each case o-' =^glc, as in the case of waves on an unlimited sheet of deep water. 
These results, which may easily be verified ab initio, were given by Kirchhoff (Lc). 

259. An interesting problem which presents itself in this connection is 
that of the transversal oscillations jof water contained in a canal of circuUiT 
section. This has not yet been solved, but it may be worth while to point 
out that an approximate determination of the frequency of the slowest mode, 
in the case where the free surface is at the level of the axis, can be effected 
by Rayleigh's method, explained near the end of Art. 168. 

If we assume as an 'approximate type' that in which the free surface 
remains always plane, making a small angle (say) with the horizontal, Jt 
appears, from Art. 72, 3<^, that the kinetic energy T is given by 

22' = (t - I) />«*^'. (1) 

where a is the radius, whilst for the potential energy V we have 

27 = |<7pa»e^ (2) 

If we assume that ac cos {at + c), this gives 

-^-ig^i <') 

whence a = 1*169 {gjay*. 

In the case of a rectangular section of breadth 2a, and depth a, the speed 
is given by Art. 257 (4), where we must put k = 7r/2a from Art. 178, and 
A == a. This gives 

a« = \tt tanh Jtt . ^ , (4) 

♦ Rayleigh finds, as a closer approximation, <r = 1*1644 (gr/a)'; see PhU, Mag. (5), t. xlviii 
p. 666 (1899) [Papers, t. iv. p. 407]. 



438 Surface Waves [chap, ix 

or a = 1*200 (g/o) • The frequency in the actual problem is less, since the 
kinetic energy due to a given motion of the surface is greater, whilst the 
potential energy for a given deformation is the same. Cf . Art. 45. 

260. We may next consider the free oscillations of the water included 
between two transverse partitions in a uniform horizontal canal. Before 
proceeding to particular cases, we may examine for a moment the nature of 
the analytical problem. 

If the axis of x be parallel to the length, and the origin be taken in one 
of the ends, the velocity-potential in any one of the fundamental modes 
referred to may, by Fourier's Theorem, be supposed expressed in the form 

<f>= {Po + Pi cos kx-{- P^ cos 2hx + . . • + P, cos skx 4- . . . ) cos (a< + c), 

(1) 

where k = tt/J, if I denote the length of the compartment. The coefficients 
P, are here functions of y, z. If the axis of z be drawn vertically upwards, 
and that of y be therefore horizontal and transverse to the canal, the forms 
of these functions, and the admissible values of a, are to be determined from 
the equation of continuity 

VV = (^, (2) 

with the conditions that ^ ~ ^ (^) 

at the sides, and that a*<f> = 9 ^ (4:) 

at the free surface. Since d<f>ldx must vanish for a; = and x = l, it follows 
from known principles* that each term in (1) must satisfy the conditions 
(2), (3), (4) independently; viz. we must have 

gap gap 

8^' + ll-' - '•*'^' = « (5) 

-*t W = « («) 

at the lateral boundary, and 



-'P' = 9^-W (^> 



at the free surface. 



The term Pq gives purely transverse oscillations such as have been 
discussed in Art. 258. Any other term Pg cos shx gives a series of fundamental 
modes with s nodal lines transverse to the canal, and 0, 1, 2, 3, ... nodal lines 
parallel to the length. - 

* See Stokes, "On the Critical Values of the Sums of Periodic Series,** Catnb. Trans, t. viii. 
(1847) [Papers, t. i. p. 236], 



259-260] Waves in Uniform Caned 439 

It will be sjifficient for our purpose to consider the term Pj cos kx. It is 
evident that the assumption 

^ = Pj cos fee . cos ((rf -f €), (8) 

with a proper form of Pj and the corresponding value of a determined as 
above, gives the velocity-potential of a possible system of standing waves, of 
arbitrary wave-length 27r/i, in an unlimited canal of the given form of 
section. Now, as explained in Art. 229, by superposition of two properly 
adjusted systems of standing waves of this type we can build up a system of 
progressive waves 

<f> = P^C08{kxTat) (9) 

We infer that progressive waves of simple-harmonic profile, of any assigned 
wave-length, are possible in an infinitely long canal of any uniform section. 

We might go further, and assert the possibility of an infinite number of 
types, of any given wave-length, with wave-velocities ranging from a certain 
lowest value to infinity. The types, however, in which there are longitudinal 
nodes at a distance from the sides are from the present point of view of 
subordinate interest. 

Two extreme cases call for special notice, viz. where the wave-length is 
very great or very small compared with the dimensions of the transverse 
section. 

The most interesting types of the former class have no longitudinal nodes, 
and are covered by the general theory of 'long ' waves given in Arts. 169, 170. 
The only additional information we can look for is as to the shapes of the 
wave-ridges in the direction transverse to the canal. 

In the case of relatively short waves, the most important type is one in 
which the ridges extend across the canal with gradually varying height, 
and the wave-velocity is that of free waves on deep water as given by 
Art. 229 (6). 

There is another type of short waves which may present itself when the 
banks are inclined, and which we may distinguish by the name of 'edge- 
waves,' since the amplitude diminishes exponentially as the distance from the 
bank increases. In fact, if the amplitude at the edges be within the limits 
imposed by our approximations, it will become altogether insensible at 
a distance whose projection on the slope exceeds a wave-length. The wave- 
velocity is less than that of waves of the same length on deep water. It 
does not appear that the type of motion here referred to is very important. 

A general formula for these edge- waves has been given by Stokes*. 
Taking the origin in one edge, the axis of z vertically upwards, and that of y 

* "Report on Recent Researohes in Hydrodynamics," Brit. Ass. Rep. 1846 [Papers, t. i. 
p. 167]. 



440 Surface Waves [chap, ix 

transverse to the canal, and treating the breadth as relative^ infinite, the 
formula in question is 

4> = jye-»(»««^-*«^^' cos Jk (a; - cf), (10) 

where j3 is the slope of the bank to the horizontal, and 

c=(|8m/3)* (11) 

The reader will have no difficulty in verifying this result. 

261. We proceed to. the consideration of some special cases. We shall 
treat the question as one of standing waves in an infinitely long canal, or in 
a compartment boimded by two transverse partitions whose distance apart is 
a multiple of half the arbitrary wave-length (27r/A;), but the investigations 
can easily be modified as above so as to apply to progressive waves, and we 
shall occasionally state results in terms of the wave- velocity. 

1°. The solution for the case of a rectangular section, with horizontal bed and vertical 
sides, could be written down at once from the results of Arts. 190, 257. The nodal lines 
are transverse and longitudinal, except in the case of a coincidence in period between two 
distinct modes, when more complex forms are possible. This will happen, for instance, in 
the case of a square tank. 



o 



2^. In the case of a canal whose section consists of two straight lines inclined at 45 
to the vertical we have, first, a type discovered by Kelland* : viz. if the axis of x coincide 
with the bottom line of the canal, 

ky Icz 
(f) =A cosh -t| cosh j^ cos kx . cos ((ri +c) (1) 

This evidently satisfies V*0 =0, and makes 

|=4t' (2) 

for y = ±2, respectively. The surface-condition (Art. 260 (4)) then gives 

'^•=3*-^|' <') 

where k is the height of the free surface above the bottom line. If we put <r =fcc, the wave- 
velocity c is given by 

'="=72ifc*''"\^ <*> 

where k =27r/X, if X be the wave-length. 
When hfk is small, this reduces to 

c={\gh)K (5) 

in agreement with Art. 170 (13), since the mean depth is now denoted by \h. 
When, on the other hand, hjX is moderately large, we have 

c2__?_ (6) 



ft 



* " 



On Waves," Trans, R. 8. Edin, t. xiv. (1839). 



260-261] Canal of Triangvlar Section 441 

The formula (1) indioatoe now a rapid increase of amplitude towards the sides. We 
have here, in fact, an instance of * edge- waves,' and the wave-velocity agrees with that 
obtained by patting /9=45^ in Stokes' formula. 

The remaining types of oscillation which are symmetrical with respect to the medial 
plane y =0 are given by the formula 

<^ =0 (cosh ay cos jSz +cos fiy cosh az) cos lex . cos (o-^ +c) (7) 

provided a, /9, o- are properly determined. This evidently satisfies (2), and the equation of 
continuity gives 

a«-i9*=it* (8) . 

The surface-condition, Art. 260 (4), to be satisfied for z =%, requires 

a^ cosh ah=ga sinh aA, o-' cos fih= -g^mnffh (9) 

Hence aAtanhaA-i-/9^tan/3A:=0 (10) 

The values of a, /3 are determined by (8) and (10), and the corresponding values of a are 
then given by either of the equations (9). If, for a moment, we write 

aj=aA, y=fih, •• (11) 

the roots are given by the intersections of the curve 

a;tanhx+t/tan^=0, (12) 

whose general form can be easily traced, with the hyperbola 

x«-y«=ibW (13) 

There are an infinite number of real solutions, with values of fih lying in the second, fourth, 
sixth, . . . quadrants. These give respectively 2, 4, 6, ... longitudinal nodes of the free 
surface.' When hjX is moderately large, we have tanh<iA = l, nearly, and /3^ is (in the 
simplest mode of this class) a little greater than \ir. The two longitudinal nodes in this 
case approach very closely to the edges as X i^ diminished, whilst the wave- velocity becomes 
practically equal to that of waves of length X on deep water. As a numerical example, 
assuming ffh = \*\x ^n-, we find 

ah = 10-910, kh = 10-772, c = 10064 (1^ . 

The distance of either nodal line from the nearest edge is then -12^. 

We may next consider the asymmetrical modes. The solution of this type which is 
analogous to Kelland's was noticed by Greenhill (Z.c). It is 

(f) —A sinh -^sinh -t^ cos kx . cos (vt +€), (14) 

with ^*=^°^**^^ ^^^' 

When kh is small, this makes (r*=g/h, so that the * speed' is very great compared with 
that given by the theory of 'long' waves. The oscillation is in fact mainly transversal, 
with a very gradual variation of phase as we pass along the canal. The middle line of the 
surface is of course nodal. When kh is great, we get * edge- waves,' as in the case of 
Eelland's solution. 

The remaining asymmetrical oscillations are given by 

<^ =A (sinh ay sin fiz -Hsin /3y sinh az) oob kz , ooe {ai -{-t) (16) 

This leads in the same manner as before to 

a«-i9*=*« (17) 



442 



SurfoM Waves 



[chap. IX 



and o-' sinh ak^ga cosh oA, cr' sin ^=gff oos fih, (18) 

whence oAcoth ah=&hcotfih (19) 

There are an infinite number of solutions, with values of /3A in the third, fifth, seventh, . . . 
quadrants, giving 3, 5, 7, ... longitudinal nodes, one of which is central. 

3<*. The case of a canal with plane sides inclined at 60° to the vertical has been treated 
by Macdonald*. He has discovered a very comprehensive type, which may be verified as 
follows. 



The assumption 



(f) =sP cos kx , cos {(rl +c). 



(20) 



where P=.4 cosh ib+fi sinh Jb+ cosh ^?^^C7 cosh ^+ 1) sinh ^^ (21) 

evidently satisfies the equation of continuity ; and it is easily shewn that it makes 



for y = ±V3«, provided C=2A, D= -2B 

The surface-condition, Art. 260 (4), is then satisfied, provided 

-f (A cosh kh +B sinh kk) =A sinh kk+B cosh kh, 

— r ( A cosh IT - -o smh 
gk\ 2 

The former of these is equivalent to 
A =H ( cosh ^A - -T sinh kh 
and the latter then leads to 



(22) 



inh -^ j =A si 



. ,kh „ ,kh 
smh -^ - B cosh -5- . 



(23) 



y B=H(^coBhkh-smhkh\ (24) 



kh 

coth3 2-+l=0. 



\gk) gk 
Also, substituting from (22) and (24) in (21), we find 

P=H jcosh k{z-h)+^amhk(z - A)| 



(26) 



+ 2H cosh ^^ jcosh k(^ + aV ^sinhifc^l + h\\. . . .(26) 



The equations (25) and (26) were arrived at by Maodonald, by a different process. 
The surface- value of P is 

Uh -« 



-( 



1 + 2 oo8h M (coBb^-^ - ^ sinh ^)| . 



(27) 



The equation (25) is a quadratic in (r*/gk. In the case of a wave whose length (Zv/k) 
is great compared with h, we have 



nearly, and the roots of (25) are then 



,, 3*A 2 
*"**'' -2" =3iWi' 



5=*^ and ^ = 1M 



(28) 



approximately. If we put a=kc, the former result gives c^ =igh, in accordance with the 
usual theory of Uong' waves (Arts. 169, 170). The formula (27) now makes P = 3H, 
approximately; •this is independent of y, so that the wave-ridges are nearly straight. The 

♦ "Waves in Canals," Proc. Land. Math. Soc. t. xxv. p. 101 (1894). 



261-262] Canal of Triangtdar Section 443 

second of the roots (28) makes c' =g/h, giving a much greater wave- velocity ; but the con- 
siderations adduced above shew that there is nothing paradoxical in this. It will be found 
on examination that the cross-sections of the waves are parabolic in form, and that there 
are two nodal lines parallel to the length of the canal. The period is, in fact, almost 
exactly that of the symmetrical transverse oscillation discussed in Art. 258. 

When, on the other hand, the wave-length is short comp>ared with the transverse 
dimensions of the canal, kh is large, and coth §M = 1, nearly. The roots of (26) are then 

iJ='-<^p=4 (2«) 

approximately. The formef result makes P=H, nearly, so that the wave-ridges are 
straight, experiencing only a slight change of altitude towards the sides. The speed, 

o" = {gk)\ is exactly what we should expect from the general theory of waves on relatively 
deep water. 

If in this case we transfer the origin to one edge of the winter-surface, writing z+h for z, 
and y - JSh for y, and then make h infinite, we get the case of a system of waves travelling 
parallel to a shore which slopes downwards at an angle of 30"^ to the horizon. The result is 

<^ =H {e** +e-** <^*«'+" ,3g-4*(V8y-»)^ cos ib; . cos {at +e) (30) 

where c = (g/k)^. This admits of immediate verification. At a distance of a wave-length 
or so from the shore, the value of <^, near the suHace, reduces to 

• <f>=H^*coBkx.coB{at+€), (31) 

practically, in conformity with Art. 228. Near the edge the elevation changes sign, there 
being a longitudinal node for which 



^ky=\og,2, (32) 



ory/X=127. 



The second of the two roots (29) gives a system of edge-waves, the results being equi- 
valent to those obtained by making /3=30° in Stokes* formula. 



OsdUations of a Spherical Mass of Liquid. 

262. The theory of the gravitational oscillations of a mass of liquid 
about the spherical form is due to Kelvin*. 

Taking the origin at the centre, and denoting the radius vector at any 
point of the surface by a + f , where a is the radius in the imdisturbed state, 
we assume 

C'^^L, (1) 

1 

where 2|^ is a surface-harmonic of integral order n. The equation of con- 
tinuity V Y = is satisfied by 

OO Mil 

<f>-^^~Sn, (2) 

1 »n 

* Sir W. Thomson, "Dynamical Problems regarding Elastic Spheroidal SbeUs and Spheroids 
of Incompressible Liquid,'* Phil Trana, 1863 [Papers, t. iii. p. 384]. 



444 Surface Waves [oHAPt ix 

where /S^ is a surface-harmonic, and the kinematical coadition 



3? _ ^ /o\ 



to be satisfied when r ^ a^ gives 



t— j»- • <«> 

The gravitation-potential at the free surface is (see Art. 200) 

4ir7^»_£4iryP? (5) 

where y is the gravitation-constant. Putting 

g = ^7Tyf>a, r « a -f- Sf„, 

we find Q, = const. + fl'S -k — ti Sn (6) 

1 Zii -f- 1 

Substituting from (2) and (6) in the pressure-equation 

P = ^ _ £j + const., : (7) 

p 01 

we find, since p must be constant over the surface, 

dSn 2 (n - 1) .. 

-ar=-2;rTr^^ <^^ 

Eliminating iS, between (4) and (8), we obtain 

dKn I 2n (n - 1) g y _ ,g. 

This shews that ^, oc cos (a„/. + e), where 

2n (n — 1) g 



= 



2n + 1 o* 



(10) 



For the same density of liquid, g ooa, and the frequency is therefore 
independent of the dimensions of the globe. 

The formula makes a^ = 0, as we should expect, since in the deformation 
expressed by a surface-harmonic of the first order the surface remains 
spherical, and the period is therefore infinitely long. 

''For the case n « 2, or an ellipsoidal deformation, the length of the 
isochronous simple pendulum becomes fa, or one and a quarter times the 
earth's radius, for a homogeneous liquid globe of the same mass and diameter 
as the earth; and therefore for this case, or for any homogeneous liquid globe 
of about 5^ times the density of water, the half-period is 47 m. 12 s." 



262-263] 



Oscillations of a Liquid Globe 



445 



"A steel globe of the same dimensionB, without mutual gravitation of its 
parts, could scarcely oscillate so rapidly, since the velocity of plane waves of 
distortion in steel is only about 10,140 feet per second, at which rate a 
space equal to the earth's diameter would not be travelled in less than 
Ih. 8m. 40s.*" 




When the surface oscillates in the form of a zonal harmonic spheroid of the second 
order, the equation of the lines of motion is xw*= const., where w denotes the distance of 
any point from the axis of symmetry, which is taken as axis of x (see Art. 95 (11)). The 
forms of these lines, for a series of equidistant values of the constant, are shewn in the 
figure. 

263. This problem may also be treated very compactly by the method 
of 'normal co-ordinates ' (Art. 168). 

The kinetic energy is given by the formula 



-ip//^ 



dr 



dS, 



(11) 



where hS is an element of the surface r ^ a. Hence, when the surface 
oscillates in the form r^a-{-^^, we find, on substitution from (2) and (4), 



^ - »?//^ 



^dS. 



(12) 



* Sir W. Thomson, Lc. The exact theory of the vibrations of an elastic sphere gives, for the 
slowest oscillation of a steel globe of the dimensions of the earth, a period of 1 h. 18 m. See a 
paper "On the Vibrations of an Elastic Sphere," Proc Lond. Math, 8oc. t. xiii. p. 212 (1882). 
The vibrations of a sphere of incompressible substance, under the joint influence of gravity and 
elasticity, have been discussed by Bromwich, Proc. Lond. Math. Soc. t. xxx. p. 98 (1898). Th& 
influence of compressibility is examined by Love, Some Problems of Oeodynamica (Adams Prize 
Essay), Cambridge, 1911, p. 126. 



446 Surface Waves [chap, ix 

To find the potential energy, we may suppose that the external surface 
is constrained to assume in succession the forms r = a-\' dt,^, where varies 
from to 1. At any stage of this process, the gravitation potential at the 
surface is, by (6), 

ii = const. + ?^^</fl^„ ...(13) 

Hence the work required to add a film of thickness ^,,80 is 

eS0.^-^^gpSSi„*dS (14) 

Integrating this from 6 = to 6 = 1, we find 

V == :^^gpSnn'dS (15) 

The results corresponding to the general deformation (1) are obtained by 
prefixing the sign S of summation with respect to n, in (12) and (15); since 
the terms involving products of surface-harmonics of different orders vanish, 
by Art. 87. 

The fact that the general expressions for T and V thus reduce to sums 
of squares shews that any spherical-harmonic deformation is of a 'normal 
type.' Also, assuming that f „ oc cos (cTnt + c), the consideration that the 
total energy T -\- V must be constant leads us again to the result (10). 

In the case of the forced oscillations due to a disturbing potential 
Q' cos (at + €) which satisfies the equation V*i2' = at all points of the fluid, 
we must suppose Q' to be expanded in a series of solid harmonics. If f„ be 
the equilibrium-elevation corresponding to the term of order n, we have, by 
Art. 168 (14), for the forced oscillation, 

^" = i^:^^;V^*f»' .....(16) 

where -tj is the imposed speed, and a„ that of the free oscillations of the same 
type, as given by (10). 

The numerical results given above for the case n = 2 shew that, in a non- 
rotating liquid globe of the same dimensions and mean density as the earth, 
forced oscillations having the characters and periods of the actual lunar and 
solar tides would practically have the amplitudes assigned by the equilibrium- 
theory. 

264. The investigation is easily extended to the case of an ocean of any 
uniform depth, covering a symmetrical spherical nucleus. 

Let b be the radius of the nucleus, a that of the external surface. The surface-form 
being 

r=a + j:*Cn (1) 



263-264] Ocean of Uniform Depth 447 

we aasiime, for the velocity-potential, 

« = {(^+l)5-„ + ^^i}^" (2) 

where the coefficients have been adjusted so as to make d<f)/dr=0 for r=b. 

The condition that ^ ~S^ '^' ^^' 

,<„=ig.™ t- -«(-.! {©■-(ID I (« 

For the gravitation-potential at the free surface (1) we have 

"- . 3r ^i2n + l'"' ^^^ 

where po is the mean density of the whole mass. Hence, putting g =|9rypoa» we find 

■ Q=oonBt.+ysr(i-2innp-,)f- <«> 

The pressure-condition at the free surface then gives 

The elimination of 8n between (4) and (7) leads to 

^+(r»V.=0, (8) 

where «r.« = - iM—W^A 3 _pW _ 

If p =po, we have o-^ =0 as we should expect. When p>po the value of o-i is imaginary; 
the equilibrium configuration in which the external surface of the fluid is concentric with 
the nucleus is then imstable. (Cf. Art. 200.) 

If in (9) we put 6=0, we reproduce the result of the preceding Art. 'If, on the other 
hand, the depth of the ocean be small compared with the radius, we find, putting b=a-h, 
and neglecting the square of h/a. 



...=n(n.l,(l-2-A.£)g (10) 



provided n be small compared with a/h. This agrees with Laplace's result, obtained in a 
more direct manner in Art. 200. 

But if n be comparable with a/h, we have, putting n =ka, 

so that (9) reduces to <r* -gk tanh kh, (II) 

as in Art. 228. Moreover, the expression (2) for the velocity-potential becomes, if we 
write r=a+z, 

=<l>i cosh A; (z +h)^ (12) 

where 0^ is a function of the co-ordinates in the surface, which may now be treated as 
plane. Cf. Art. 257. 



448 Surface Waves [chap, ix 

The formulae for the kinetio and potential energies, in the genera] case, are easily found 
by the same method as in the preceding Art. to be 



and 



y=i9pj::{l-^,^)SiC,^d8. (14) 



The latter result shews, again, that the equilibrium configuration is one of minimum 
potential energy, and therefore thoroughly stable, provided p<po* 

In the case where the depth Ls relatively small, whilst n is finite, we obtain, putting 
b=a-h. 



''=*T^2:,-(i^)j/f.«<i5. (15) 



whilst the expression for F is of course unaltered. 

If the amplitudes of the harmonics (n be regarded as generalized co-ordinates, the 
formula (15) shews that for relatively small depths the 'inertia-coefficients* vary inversely 
as the depth. We have had frequent illustrations of this principle in our discussions of 
tidal waves. 

CapiUarity. 

265. The part played by Cohesion in certain cases of fluid motion has 
long been recognized in a general way, but it is only within comparatively 
recent years that the question has been subjected to exact mathematical 
treatment. We proceed to give some account of the remarkable investi- 
gations of Kelvin and Bayleigh in this field. 

It is beyond our province to discuss the physical theory of the matter*. 
It is sufficient, for our purpose, to know that the free surface of a liquid, or, 
more generally, the common surface of two fluids which do not mix, behaves 
as if it were in a state of uniform tension^ the stress between two adjacent 
portions of the surface, estimated at per imit length of the common boundary- 
line, depending only on the nature of the two fluids and on the temperature. 
We shall denote this 'surface-tension,' as it is called, by the symbol T^. The 
* dimensions' of Ti are MT"* on the absolute system of measurement. Its 
value in c.g.s. units (dynes per linear centimetre) appears to be about 74 for 
a water-air surface at 20° C.f ; it diminishes somewhat with rise of tem- 
perature. The corresponding value for a mercury-air surface is about 540. 

• For this, see Maxwell, Encyc. Briiann. Art. "Capillary Action" [Papers, Cambridge, 1890, 
t. ii. p. 641], where references to the older writers are given. Also, Rayleigh, *'0n the Theory 
of Surface Forces," PhU, Mag. (6), t. xxx. pp. 286, 466 (1890) [Papers, t. iii. p. 397]. 

t Rayleigh, "On the Tension of Water-Surfaces, Clean and Contaminated, investigated by 
the method of Ripples," Phil Mag. (6), t. zzz. p. 386 (1890) [Papers, t. iii. p. 394]; Pedersen, 
Pha. Trans. A, t. ccvii. p. 341 (1907); Bohr, PhU. Trans. A, t. ccix. p. 281 (1909). 



264-266] Inflitence of Cohesion 449 

An eqiuvalent statement is that the 'free' energy of any system, of 
which the surface in question forms part, contains a term proportional to the 
area of the surface, the amount of this 'superficial energy ' (as it is usually 
termed) per unit area being equal to Ti*. Since the condition of stable 
equilibrium is that the free energy should be a minimum, the surface tends 
to contract as much as is consistent with the other conditions of the problem. 

The chief modification which the consideration of surface-tension will 
introduce into our previous methods is contained in the theorem that the 
fluid pressure is now discontinuous at a surface of separation, viz. we have 



p-^'-'^^ihi)' 



where y, p' are the pressures close to the surface on the two sides, and Ri, J?, 
are the principal radii of curvature of the surface, to be reckoned negative 
when the corresponding centres of curvature lie on the side to which the 
accent refers. This formula is readily obtained by resolving along the normal 
the forces acting on a rectangular element of a superficial film, boimded by 
lines of curvature; but it seems unnecessary to give here the proof, which 
may be found in most modern treatises on Hydrostatics. 

266. The simplest problem we can take, to begin with, is that of waves 
on a plane surface forming the common boundary of two fluids at rest. 

If the origin be taken in this plane, and the axis of y normal to it, the 
velocity-potentials corresponding to a simple-harmonic deformation of the 
common surface may be assumed to be 

<f> = Ce^^ cos kx . cos (ai + €), 1 .- . 

^' = C'e^^^ cos kx . cos (at + c), j 

where the former equation relates to the side on which y is negative, and 
the latter to that on which y is positive. For these values satisfy V^ = 0, 
V^' = 0, and make the velocity zero f or y = T oo , respectively. 

The corresponding displacement of the surface in the direction of y will 
be of the type 

7) = a COS kx . ain (ai -\- €); (2) 

and the conditions that 

drj d(f> d<f> 

dt^ dy" dy * 
for y = 0, give 

aa = - JtC = AC' (3) 

* The distinction between *free* and * intrinsic' energy depends on thermo-dynamical 
principles. In the case of changes made at constant temperature with free communication of 
heat, it is with the 'free* energy that we are concerned. 

L. H. 29 



450 Surface Waves [chap, ix 

If, for the moment, we ignore gravity, the variable part of the pressure 
is given by 

- = ~- = — =— e*" cos kx . sm (at -j- €h 
ot k 



^ = -^ = e-*vcos kx , sin (at + €). 

p dt k 



(4) 



To find the pressure-condition at the common surface, we may calculate 
the forces which act in the direction of y on a strip of breadth 8x, The 
fluid pressures on the two sides have a resultant (p' —p) Sx, and the difference 
of the tensions parallel to y on the two edges gives S (Tidrj/dx). We thus 
get the equation 

P-P' + T^l^» = 0. (5) 

to be satisfied when y = approximately. This might have been written 
down at once as a particular case of the general surface condition (Art. 265). 
Substituting in (5) from (2) and (4), we find 

^« = /_^^,, (6) 

which determines the speed of the oscillations of wave-length 27r/i. 

The energy of motion, per wave-length, of the fluid included between two planes 
paraUel to xy, at unit distance apart, is 

^■»'/:[*ii.*-»''/:[*'iL'- '" 

If we assume 17 =a cos kx « (8) 

where a depends on t only, and therefore, having regard to the kinematical conditions, 

= - k-^a^ cos kx, <!>' =k-^ de-*» cos fee, (9) 

we find T = J (p +p') ^-l a« . X (10) 

Again, the energy of extension of the surface of separation is 

-'■■/:^(i)r^-''.-»''-/:®"^ '••••'■■' 

Substituting from (8), this gives 

7 =iTiitV . X (12) 

To find the mean energy, of either kind, per unit area of the common surface, we must 
omit the factor X. 

If we assume that a occos {at +c), where o- is determined by (6), we verify that the 
total energy T + V is constant. Conversely, if we assume that 

17=2 (acosibx+^sinArx), (13) 

it is easily seen that the expressions for T and V will reduce to sums of squares of a, 4 
and a, /9, respectively, with constant coefficients, so that the quantities a, fi are * normal 
co-ordinates.' The general theory of Art. 168 then leads independently to the formula (6) 
for the speed. 

By compounding two systems of standing waves, as in Ajrt. 229, we obtain 

a progressive wave-system 

7y = a cos (fee =F at), (14) 



266] 



Capillary Waves 



451 



travelling with the velocity 

"" " A "" V+ p') ' 
or, in terms of the wave-length, 



(15) 



c=fMi,y.A-i (16) 

The contrast with Art. 229 is noteworthy ; as the wave-length is diminished, 
the period diminishes in a more rapid ratio, so that the wave-velocity 
increases. * 

Since c varies as A ~ % the group-velocity is, by Art. 236 (3), 



^ = "-^5A = ^''- 



(17) 



The verification of the relation between group- velocity and transmission of energy is of 

some interest. Taking 

fj=acosk(ct -x)j (18) 

we find that the total energy per unit area of the surface is 

l(p+p')*cV+JT,ifcV=i(pV)^'«*» (1») 

by ( 10), ( 12 ). The mean rate at which work is done by fluid pressure at a plane perpendicular 
to X is found by a calculation similar to that of Art. 237 to be 

i (p +p') ikc»a* , (20) 

The rate at which surface tension does work at such a plane is 

Ti ^ ^ = T'l IcHa^ sin* it (c< - x), 

the mean value of which is 

i7'iifc«ca«=i (p + p') itc»a* (21) 

If we add this to (20), and divide by the second member of (19), the quotient is f c, in 
agreement with (17). 

The fact that the group- velocity for capillary waves exceeds the wave- 
velocity helps to explain some interesting phenomena to be referred to later 
(Arts. 271, 272). 

For numerical illustration we may take the case of a free water-surface ; 
thus, putting /> = 1, p' = 0, Ti = 74, we have the following results, the units 
being the centimetre and second*. 



Wave-length 


Wave- velocity 


Frequency 


•50 
•10 
•05 


30 
68 
96 


61 

680 

1930 



* Cf. Sir W. Thomson, Papers, t. iii. p. 620. 

The above theory gives the explanation of the crispations observed on the surface of water 
contained in a finger-bowl set into vibration by stroking the rim with a wetted finger. It is to 
be observed, however, that the frequency of the capillary waves in this experiment is double that 
of the vibrations of the bowl; see Rayleigh, "On Maintained Vibrations/' PhU. Mag. (5), t. xv. 
p. 229 (1883) {Papers, t. ii. p. 188; Theory of Sound, 2nd ed., c. xx.]. 

29—2 



452 Surface Waves [chap* ix 



( -jT — flr J a COS hx . sin {eft + c), 



(1) 



267. When gravity is to be taken into account, the common surface, in 
equilibrium, will of course be horizontal. Taking the positive direction of y 
upwards, the pressure at the disturbed surface will be given by 

^ = -^ - gy ^ - (^ + gj a COB kx. sin {at + c), 
approximately. Substituting in Art. 266 (5), we find 

.•='^;,*+,^ ™ 

Putting <7 = ic, we find, for the velocity of a train of progressive waves, 

c2==^_-/>;f4. Jl^i^J-ii-Vf + riV (3) 

p-hp' k^ p + p' l-{-8\k^ J' ^ ' 

where we have written 

^ = ., -^^-, = r (4) 

p p-- p 

In the particular cases of Tj = and flr = 0, respectively, we fall back on 
the results of Arts. 232, !<>, and 266. 

There are several points to be noticed with respect to the formula (3). 
In the first place, although, as the wave-length {^jk) diminishes from oo to 
0, the speed {a) continually increases, the wave-velocity, after falling to 
a certain minimum, begins to increase again. This minimum value (c,„, say) 
is given by 

cj = } ^* . 2 ((^r )t (5) 



(6) 



and corresponds to a wave-length 

^=^-^/(7> 

In terms of A^ and c^ the formula' (3) may be written 

5=i(^T") (') 



'm >'*m 



shewing that for any prescribed value of c, greater than c^, there are two 
admissible values (reciprocals) of A/A^j. For example, corresponding to 

-= 1-2 1-4 1-6 1-8 20 

we have 

2-476 3-646 4-917 6*322 7-873 

-404 -274 -203 -158 127, 



*m 



* The theory of the minimum wave-velocity, together with most of the substance of Arts. 266, 
267, was giveif by Sir W. Thomson, '*Hydrokinetic Solutions and Observations," Phil. Mag, (4), 
t. xlii. p. 374 (1871) [Baltimore Lectures, p. 598]; see also NcUure, t. v. p. 1 (1871). 



267] Waves under Gravity a7id Capillarity 453 

to which we add, for future reference, 

sin-i ^ = 56°26' 45°35' 38°41' 33°45' 30°. 
c 

For sufficiently large values of A the first term in the formula (3) for c* 
is large compared with the second; the force governing the motion of the 
waves being mainly that of gravity. On the other hand, when A is very 
small, the second term preponderates, and the motion is mainly governed by 
cohesion, as in Art. 266. As an indication of the actual magnitudes here in 
question, we may note that if A/A^ > 3, the influence of cohesion on the 
wave-velocity amounts only to about 5 per cent., whilst gravity becomes 
relatively ineffective to a like degree if A/A^ < J. 

It has been proposed by Kelvin to distinguish by the name of 'ripples' 
waves whose length is less than An^. 

The relative importance of gravity and cohesion, as depending on the value of X, may 
be traced to the form of the expression for the potential energy of a deformation of the 
type 

Tf=aoo8kx (8) 

The part of this energy due to the extension of the bounding surface is, per unit area, 

'-^ (9) 

whilst the part due to gravity is 

i9{p-p')a* (10) 

As X diminishes, the former becomes more and more important compared with the latter. 

For a water-surface, using the same data as before, with (7=981, we find from (5) 
and (6) 

Xm = l-73, c„=23-2, 

the units being the centimetre and the second. That is to say, roughly, the minimum 
wave- velocity is about nine inches per second, or *45 sea-miles per hour, with a wave- 
length of two-thirds of an inch. Combined with the numerical results already obtained, 
this gives, 

for c= 27-8 32-6 371 41-8 46-4 



"I •' 



,, , , , .3 6-3 8-6 10-9 13-6 

the values X = ^ ,^^ .^^ .3^ .37 22 



in centimetres and seconds. 

If we substitute from (7) in the general formula (Art. 236 (3)) for the 
group-velocity, we find 

^=«-*?A-«(>-jJ!:;J!:D <") 

Hence the group-velocity is greater or less than the wave-velocity, according 
as A $ Ani. For sufficiently long waves the group-velocity is practically equal 
to Jc, whilst for very short waves it tends to the value f c*. 

* Cf. Rayleigh, U.cc. anU p. 372. 



454 



Surface Waves 



[chap. IX 



The relations between wave-length and wave-velocity are shewn 
graphically in the annexed figure, where the dotted curves refer to the 




cases where gravity and capillarity act separately, whilst the full curve 
exhibits the joint effect. As explained in Art. 236, the group-velocity is 
represented by the intercept made by the tangent on the axis of ordinates. 
Since two tangents can be drawn to the curve from any point on this axis 
(beyond a certain distance from 0), there are two values of the wave-length 
corresponding to any prescribed value of the ^rroup-velocity Z7. These two 
values of A coincide when JJ has a certain (minimum) value, indicated by 
the point where the tangent to the curve at the point of inflexion cuts Oc \ 
and it may be easily shewn that we then have 

^=y^(3 + 2V3) = 2-542, 

where c^ is the minimum t^^avc-velocity as above. 



V = -7670 



A further consequence of (2) is to be noted. We have hitherto tacitly supposed that 
the lower fluid is the denser (t.e. p>p\ as is indeed necessary for stability when Tj is 
neglected. The formula referred to shews, however, that there is stability even when 
p<p\ provided 

'<''yV))* '''\ 

t.«. provided X be less than the wave-length Xm of minimum velocity when the denser fluid 



267-268] Group-Vdocity 455 

is below. Hence in the case of water above and air below the maximum wave-length con- 
sistent with stability is 1*73 cm. If the fluids be included between two parallel vortical 
walls, this imposes a superior lin^it to the admissible wave-length, and we learn that there 
is stability (in the two-dimensional problem) provided the interval between the walls does 
not exceed -86 cm. We have here an explanation, in principle, of a familiar experiment in 
which water is retained by atmospheric pressure in an inverted tumbler, or other vessel, 
whose mouth is covered by a gauze with sufficiently fine meshes *. 

268. We next consider the case of waves on a horizontal surface forming 
the common boundary of two parallel currents U, ZJ'f. 

If ,we apply the method of Art. 233, we find without difficiilty that the 
condition for a stationary wave-profile is now 

pV^ + p'U'^ = ^^{p-p') + kT„ (1) 

the last term being due to the altered form of the pressure-condition which 
has to be satisfied at the surface. 



This may be written 

'P U + P'U' ]' ^l P__-P' + . *Zi_ _ PP' m _ uy (2) 
p + p' 1 k-p + p'.+ p + p' (p + /.')*^ '' "^' 



(' 



The relative velocity of the waves, which is superposed on the mean 
velocity of the currents (Art. 233), is ± c, provided 

c* = Co* - 7-^.-, {V-uy, (3) ■ 

(p + p)* 

where Cq denotes the wave-velocity in the absence of currents. 

The various inferences to be drawn from (3) are much as in the Art. 
cited, with the important qualification that, since Cq has now a minimum 
value, viz. the Cn, of Art. 267 (5), the equilibrium of the surface when plane 
is stable for disturbances of all wave-lengths so long as 



U^V^\<^-±^,c^, (4) 



where $ = p'/p. 



When the relative velocity of the two currents exceeds this value, c 
becomes imaginary for wave-lengths lying between certain limits. It is 

* The case where the fluids are contained in a cylindrical tube was solved by MaxweU, 
Encyc. Briiann, Art. "Capillary Action " [Papers, t. ii. p. 585], and compared with some experi- 
ments of Duprez. The agreement is better than might have been expected when we consider 
that the special condition to be satisfied at the line of contact of the surface with the wall of the 
tube has been left out of account. 

t Cf. Sir W. Thomson, Phil Mag. (4), t. xlii. p. 368 (1871) [BaUimore Lectures, p. 590]. 



456 SarfoAie Waves [chap, ix 

evident that in the alternative method of Art. 234 the time-factor e*'* will 
now take the form e±**+*^*, where 



a=^ ., ^ .. {V-Vy-c^^^k, j8 = j^Jfc|C7-C^'|. ...(5) 



8 

The real part of the exponential indicates the possibility of a disturbance of 
continually increasing amplitude. 



For the case of air over water we have 8 — -00129, Cm =23*2 (c.s.), whence the maximum 
value of \U -'T]'\ consistent with stability is about 646 centimetres per second, or (roughly) 
12-5 sea-miles per hour*. For slightly greater values the instability will manifest itself by 
the formation, in the first instance, of wavelets of about two-thirds of an inch in length, 
which will continually increase in amplitude until they transcend the limits implied in our 
approximation. 

269. The waves due to a local impulse on the surface of still water may 
be investigated to a certain extent by Kelvin's method (Art. 241). 

Since ^ = — 9^/9y at the surface, we have 

<i = — I cos aie^^ cos hcilc. rt = I — cos kxkdh. ... (1) 

Hence to conform to (6) of Art. 241 we must put 

(f>(k) = ik/p(r (2) 

If in Art. 267 (2) we put ^' = and write, for shortness, 

Tjp^r (3) 

we have a^ = gk+ TB (4) 

Let us first suppose that capillarity alone is operative, so that 

<7« = rA!» (5) 

Since S=t2"***. S = i^'**"* («> 

wefind * = iy.^2. <^ = iA'2v^. ^W^^li ^"^^ 

The procedure of Art. 241 then gives 

1 / 4a!' \ 

''^.*prM'^l27r7^"*">' ^^^ 

The test-fraction (9) of Art. 241 is now comparable with T^tjo?, and the 
approximation therefore cannot claim great accuracy except as regards the 
earlier stages of the disturbance at any point. It appears also from (8) that 

* The wind-velocity at which the surface of water actually begins to be ruffled by the forma- 
tion of capillary waves, so as to lose the power of distinct reflection, is much less than this, and 
is determined by other causes. This question is considered later (Chapter xl). 



268-270] Waves due to a Local ImpvUe 457 

the wave-length and period at any point begin by being infinitesimal, and 
continually increase. These several circumstances are in contrast with what 
holds in the case of gravity waves (Art. 240). 

We have seen (Art. 267) that when gravity is taken into account there are 
two wave-lengths corresponding to any assigned value of the group- velocity 
V which exceeds the minimum Vq. The particular wave-lengths correspond- 
ing to given values of x and t may be found by the geometrical methods of 
Art. 241. Analytically, putting dajdh = U = xjt, they are determined by the 
real values of k satisfying the equation 

((7 + 3rP)« = 4a« (gy = ^' (^i + ri») (9) 

The approximate expression for rj will accordingly consist of two terms of 
the type (8) of Art. 241, so that we have two systems of waves superposed. 
For X < U^, Kelvin's method indicates that the disturbance is unimportant*. 

When x/VQt is sufficiently large the real solutions of (9) are 

* = i$*. * = l^. (10) 

approximately, as if gravity and capillarity were respectively alone operative. 
The conditions for the validity of Kelvin's approximation in this case, viz. 
that gt^/x and afl/T't^ should both be large, are to some extent opposed, but 
admit of being reconciled if x and t are both sufficiently great. The wave- 
length must in each case be small compared with x. 

The effect of a travelling disturbance can be written down from the general 
formulae of Art. 248. If /c^, /c, be the two wave-lengths corresponding to the 
wave- velocity c, it appears from the figure on p. 454 that if /c^ < /c„ we shall 
have Vi <c, U^> c. The result will be 



v-^/-^', [*>o/ 



(11) 



, = i_M e*'^. [x < 0] 

If we put <f> (k) = iP/pa (12) 

this will be found to agree, as an approximation, with the result of the more 
complete investigation which follows. 

270. We resume the investigation of the effect of a steady pressure- 
disturbance on the surface of a running stream, by the methods of Arts. 242, 
243, including now the effect of capillary forces. This will give, in addition 
to the former results, the explanation (in principle) of the fringe of ripples 
which is seen in advance of a solid moving at a moderate spoked through still 
water, or on the up-stream side of any disturbance in a uniform current. 

* Rayleigh, Phil Mag. (6), t. xxi. p. 180 (1911). 



458 Surface Waves [chap, ix 

Beginning with a simple-harmonic distribution of pressure, we assume 
- = - a; + j3e*«' sin fee, 5? = _ y ^_ ^ky cos fee, (1) 

the upper surface coinciding with the stream-line ^ = 0, whose equation is 

y = j8 cos fee, (2) 

approximately. At a point just beneath this surface we find, as in Art. 242 (8), 
for the variable part of the pressure, 

Po = Pp {(*c* — flr) cos fe» + /ic sin kx}, (3) 

where jli is the frictional coefficient. At an adjacent point just above the 
surface we must have 

K = ^0 + I'l ^ = j8/>{(fej2 -g- PTO cos fee -h jLtc sin kx), . .(4) 
where T' is written for T^p, This is equal to the real part of 

We infer that to the imposed pressure 

Po = C cos kx (5) 

will correspond the surface-form 

_ ^ {kc^ — g — k^T^) cos fer -- /AC sin kx .^ 

^^"^ (kc^-g- k^T'Y + |LL«c« ^^ 

Let us first suppose that the velocity c of the stream exceeds the 
minimum wave-velocity (Cn,) investigated in Art. 267. We may then write 

kc^^g-^ k^r =^r{k^K^){K^-k), (7) 

where k^, k^ are the two values of k corresponding to the wave- velocity c on 
still water; in other words, 27r//ci, 27r//c2 are the lengths of the two systems 
of free waves which could maintain a stationary position in space, on the 
surface of the flowing stream. We will suppose that k^> k^. 

In terms of these quantities, the formula (6) may be written 

^ C {k — Ki) (k2 — k) cos fer — jLt' sin kx ,^. 

^y-T' (k - K^Y (/c, - kY -I- ii!^ ' ^^^ 

where /a' = fic/T\ This shews that if /a' be small the pressure is least over 
the crests, and greatest over the troughs of the waves when k is greater 
that K2 or less than /c^, whilst the reverse is the case when k is intermediate 
to /Ci, /cg. In the case of a progressive disturbance advancing over still water, 
these results are seen to be in accordance with Art. 168 (14). 



270-271] Surface- Distiirhance of a Stream 459 

271. From (8) we can infer as in Art. 243 the effect of a pressure of 
integral amount P concentrated on a line of the surface at the origin, viz. 
we find 

(k — /Ci) (/cj — k) cos kx — fjf sin kx 



y = ^T-J 



(k - Kx)* {Kt - A)» + /x'» 

The definite integral ia the real part of 



dk (9) 



/ 



^ '^^^ (10) 



(A; -*ci)(Ka -*)-♦>' 

The dissipation-coefficient fi' has been introduced solely for the purpose of making the 
problem determinate; we may therefore avail ourselves of the slight gain in simplicity 
obtained by supposing fi' to he infinitesimal. In this case the two roots of the denomi- 
nator in (10) are 

where v= — - — . 

The integral (10) is therefore equivalent to 



._j_{r^'?^-.._r_f^.i (11) 



These integrals are of the forms discussed in Art. 243. Since k^>ki, v is positive, 
and it appears that when x is positive the former integral is equal to 

27rt6*''' + r f^ dk (12) 

and the latter to I j dk (13) 

Jo ^^Kj 

On the other hand, when x is negative, the former reduces to 

r« p — ikx 

J,FT7/* ('*) 

and the latter to -2wfc*''«* + f ^— dk (16) 

Jo ^ + «:i 

We have here simplified the formulae by putting ir =0 ajier the transformations. 

If we now discard the imaginary parts of our expressions, we obtain the results which 
immediately follow. 

When jLt' is infinitesimal, the equation (9) gives, for x positive, 

^'•y^-— "^ sin ^.x + i- (X), (16) 



and, for x negative, 



'^^ y = - -^^^ sin K,x + i- (x), (17) 

X #^2 ""■ tj^ 



1- n/ X 1 ff COS fee,, ""cosfcr,,] ,-Q. 

where F{^)= i 7 dk-\ ,— — dk\ (18) 

This function F (x) can be expressed in terms of the known functions Ci ki^x. 



460 



Surface Waves 



[chap. IX 



Si Ki^x, Ci K^x, Si /cjX, by Art. 243 (30). The disturbance of level represented 
by it is very small for values of a;, whether positive 
or negative, which exceed, say, half the greater 
wave-length (27r/ici). 

Hence, beyond some such distance, the surface 
is covered on the down-stream side by a regular 
train of simple-harmonic waves of length 2'jt/ki, and 
on the up-stream side by a train of the shorter 
wave-length 27r//c2. It appears from the numerical 
results of Art. 267 that when the velocity c of the 
stream much exceeds the minimum wave-velocity 
(Cm) the former system of waves is governed mainly 
by gravity, and the latter by cohesion. 

It is worth notice that, in contrast with the case 
of Art. 243, the elevation is now finite when x = 0, 
viz. we have 

ttTi 1 




Q 



y 



K2 



- log 

K2 — Ki K 



(19) 






This follows easily from (16) and (18). 

The figure shews the transition between the two 
sets of waves, in the case of k^ = S/c^. 

The general explanation of the effects of an 
isolated pressure-disturbance advancing over still 
water is now modified by the fact that there are ttoo 
wave-lengths corresponding to the given velocity c. 
For one of these (the shorter) the group-velocity is 
greater, whilst for the other it is less, than c. We 
can thus understand why the waves, of shorter 
wave-length should be found ahead, and those of 
longer wave-length in the rear, of the disturbing 
pressure. • 

It will be noticed that the formulae (16), (17) 
make the height of the up-stream capillary waves 
the same as that of the down-stream gravity waves ; 
but this result will be greatly modified when the 
pressure is diffused over a band of sensible breadth, 
instead of being concentrated on a mathematical 
line. If, for example, the breadth of the band do 
not exceed one-fourth of the wave-length on the 
down-stream side, whilst it considerably exceeds 
the wave-length of the up-stream ripples, as may happen with a very moderate 





271] Waves and Ripples 461 

velocity, the different parts of the breadth will on the whole reinforce one 
another as regards their action on the down-stream side, whilst on the up- 
stream side we shall have 'interference,' with a comparatively small residual 
amplitude. 

This point may be illustrated by assumiDg that the integral surface-pressure P has 
the distribution 

^' = 1^^ (20) 

which is more diffused, the greater the value of 6. 

The method of calculation will be understood from Art. 244. The result is that on the 
down-stream side 



and on the up-stream side 



op 

''=-pr(K;^)*"'**'^"*-^"- <22) 



where the terms which are insensible at a distance of half a wave-length or so from the 
origin are omitted. The exponential factors shew the attenuation due to diffusion ; this is 
greater on the side of the capillary waves, since k^>k^ . 

When the velocity c of the stream is less than the minimum wave- 
velocity, the factors of 

are imaginary. There is now no indeterminateness caused by putting ^ = 
ab initio. The surface-form is given by 



y 



P [* cosib 



The integral might be transformed by the previous method, but it is evident 
a priori that its value tends rapidly, with increasing x, to zero, on account 
of the more and more rapid fluctuations in sign of cos kx. The disturbance 
of level is now confined to the neighbourhood of the origin. For a; = we 
find 

y = ^— , - fl + - sin-i --,) (24) 

Finally we have the critical case where c is exactly equal to the minimum 
wave-velocity, and therefore kj = k^. The first term in (16) or (17) is now 
infinite, whilst the remainder of the expression, when evaluated, is finite. 
To get an intelligible result in this case it is necessary to retain the 
frictional coefficient fi. 

If we put fi =2a7*, we have 

(ifc-K)*+t>' = {ifc-(ic-i-w-ior)} {k-iK-m+iw)} (25) 

so that the integral (10) may now be equated to 

^-Pir L , ^^ ' x ^^- r u-r^ -^M (26) 

4m [J k-{K-xn +tm) J o k -(k +XS -iw) J 



462 Surface Waves [chap, ix 

The formulae of Art. 243 shew that when w is small the most important part of this 
expression, for points at a distance from the origin on either side, is 

V^- . 2 TT i> *'*. : ( 27 ) 

It appears that the surface-elevation is now given by 

TT . y = - -71 cos (/ex - Jtt) (28) 

The examination of the effect of inequalities in the bed of a stream, by 
the method of Art. 246, must be left to the reader. 

272. The investigation by Rayleigh*, from which the foregoing differs 
principally in the manner of treating the definite integrals, was undertaken 
with a view to explaining more fully some phenomena described by Scott 
Russell t and Kelvin J. 

" When a small obstacle, such as a fishing line, is moved forward slowly 
through still water, or (which of course comes to the same thing) is held 
stationary in moving water, the surface is covered with a beautiful wave- 
pattern, fixed relatively to the obstacle. On the up-stream side the 
wave-length is short, and, as Thomson has shewn, the force governing the 
vibrations is principally cohesion. On the down-stream side the waves are 
longer, and are governed principally by gravity. Both sets of waves move 
with the same velocity relatively to the water; namely, that required in 
order that they may maintain a fixed position relatively to the obstacle. 
The same condition governs the velocity, and therefore the wave-length, of 
those parts of the pattern where the fronts are oblique to the direction of 
motion. If the angle between this direction and the normal to the wave- 
front be called d, the velocity of propagation of the waves must be equal 
to Vo cos d, where v^ represents the velocity of the water relatively to the fixed 
obstacle. 

"Thomson has shewn that, whatever the wave-length may be, the 
velocity of propagation of waves on the surface of water cannot be less than 
about 23 centimetres per second. The water must run somewhat faster than 
this in order that the wave-pattern may be formed. Even then the angle Q 
is subject to a limit defined by Vq cos Q = 23, and the curved wave-front 
has a corresponding asymptote. 

"The immersed portion of the obstacle disturbs the flow of the liquid 
independently of the deformation of the surface, and renders the problem in 
its original form one of great difficulty. We may however, without altering 
the essence of the matter, suppose that the disturbance is produced by the 

♦ /.c. ariU p. 389. 

t "On Waves," Erii. Aaa. Rep. 1844. 

X I'C, ante p. 452. 



271-272] Effect of a Travelling Disturbance 463 

application to one point of the surface of a slightly abnormal pressure, such 
as might be produced by electrical attraction, or by the impact of a small 
jet of air. Indeed, either of these methods — the latter especially — gives 
very beautifid wave- patterns*." 

The character of the wave-pattern can be made out by the method 
explained near the end of Art. 256. 

If we take account of capillarity alone, the formida (19) of the Art. cited 
gives 

c* cos« » = F« = =^ , (1) 

by Art. 266, and the form of the wave-ridges is accordingly determined by 
the equation 

p = a sec2 d (2)t 

This leads to 

a; = a sec d (1 - 2 tan^ 6), y = 3a sec tan ^ (3) 

When gravity and capillarity are both regarded, we have, by Art. 267, 

c«co8»»=F» = ^ + ^' (4) 

Hence, if we put 

c„ = (4</r')*. 6 = 27r(^y, (5) 

^"^*^« co^=H6 + aJ' (^) 

where cos a = Cj^jc (7) 

The relation between p and 6 is therefore of the form 



or 



cos* ^ _ n / p a cos* a\ .„. 

^s^i^^Ucos^a"^ TP r ^^ 

2 = cos* » ± V(co8* d - cos* a) (9) 



The four straight lines for which ^ = ± a are asymptotes of the curve thus 
determined. The values of Jtt — a for several values of the ratio cjc^ have 
been given in Art. 267. 

When the ratio cjc^ is at all considerable, a is nearly equal to Jtt, and the 
asymptotes make very acute angles with the axis of x. The upper figure on 
the following page gives the part of the curve which is relevant to the 
physical problem in the case of c = lOCmJ. The ratio of the wave-lengths of 

* Rayleigh, Lc, 

t Since V is now > V, it appears from Art. 266 (20) that the constant a must be negative. 
} The necessary calculations were made by Mr H. J. Woodall. The scale of the figure does 
not admit of the asymptotes being shewn distinct from the curve. 



464 



Sfarface Waves 



[chap. IX 





272-273] Wave-Patterm 465 

the 'waves ' and the 'ripples ' in the line of symmetry is then, of course, 
very great. The curve should be compared with that which forms the basis 
of the figure on p. 427. 

As the ratio cjc^ is diminished, the asymptotes open out, whilst the two 
cusps on ^thei: side of th^.axis approach one another, coincide, and finally 
disappear*. The wave-system has then a configuration of the kind shewn in 
the lower diagram, which is drawn for the case where the ratio of the wave- 
lengths in the line of symmetry is 4 : 1. This corresponds to a = 26° 34', or 

c=M2(;„t- 

When c<Cj^^ the wave-pattern disappears. 

273. Another problem of great interest is the determination of the 
nature of the equilibrium of a cylindrical column of liquid, of circular section. 
This contains the theory of the well-known experiments of Bidone, Savart, 
and others, on the behaviour of a jet issuing under pressure from a small 
orifice in the wall of a containing vessel. It is obvious that the imiform 
velocity in the direction of the axis of the jet does not afiect the dynamics 
of the question, and may be disregarded in the analytical treatment. 

We will take first the two-dimensional vibrations of the column, the 
motion being supposed to be the same in each section. Using polar 
co-ordinates r, in the plane of a section, with the origin in the axis, we may 
write, in accordance with Art. 63, 

r* 
<f> = A — cos 80 . cos (of -i- €), (1) 

where a is the mean radius. The equation of the boundary at any instant 

will then be 

r = a+f, (2) 

sA 
where . . { .== coa 80 .^n {at + e)^ (3) 

the relation between the coefficients being determined by 

dt^ dr' • ^*^ 

for r == a. For the variable part of the pressure inside the column, close to 
the surface, we have 

^ = -^ = — (7-4 cos «& . sin (o< -i- c) (5) 

p ot 

The curvature of a curve which differs infinitely little from a circle having 
its centre at the origin is found by elementary methods to be 

* A tentative diagram shewed that they were nearly ooincident for c=2cm (a =60^). 

t The figure may be compared with the drawing, from observation, given by Scott Russell, Lc, 

L.H. 30 



466 Surface Waves [chap, ix 

or, in the notation of (2), 

n^a-a^^-^W) (®) 

Hence the surface-condition 

T 
p = -^ + const. (7) 

gives, on substitution from (5), 

cr« = «(««- 1) ^, (8)* 

For « = 1, we have a = 0; to our order of approximation the section 
remains circidar, being merely displaced, so that the equilibrium is neutral. 
For all other integral values of «, o^ is positive, so that the equilibrium is 
thoroughly stable for two-dimensional deformations. This is evident d priori, 
since the circle is the form of least perimeter, and therefore least energy, 
for given sectional area. 

In the case of a jet issuing from an orifice in the shape of an ellipse, an 
equilateral triangle, or a square, prominence is given to the disturbance of 
the type « = 2, 3, or 4, respectively. The motion being steady, the jet 
exhibits a system of stationary waves, whose length is equal to the velocity 
of the jet multiplied by the period {27t/g)'\. 

274. Abandoning now the restriction to two dimensions, we assume 

that 

(f> = (f>i cos kz . cos (of + e), (9) 

where the axis of z coincides with that of the cylinder, and <f>i is a function 

of the remaining co-ordinates x, y. Substituting in the equation of continuity, 

V«^ = 0, we get 

(Vx> - *«) ^1 = 0, (10) 

where Vj* = 3"/9x* + 9V9y*' I^ w® P^^ a; = r cos d, y = r sin d, this may be 
written 



g^+^a, +r»g^ A^i-U (11) 



This equation is of the form considered in Arts. 101, 191, except for the sign 
of Jk* ; the solutions which are finite f or r = are therefore of the type 



^1 = B/. (*r) ^'1 «0, (12) 



* For the original investigation, by the method of energy, see Rayleigh, "On the Instability 
of Jets," Proc, Lond. Math. Soc. t. z. p. 4 (1878), and "On the Capillary Phenomena of Jets," 
Proc, Boy. Soc. t. xxix. p. 71 (1879) [Papers, t. i. pp. 361, 377 ; Theory of Sound, 2nd ed. c. xx.]. 
The latter paper contains a comparison of the theory with experiment. 

f It is assumed that this wave-length is large compared with the circumference of the jet. 
Otherwise, the formula (18) must be employed, with <r = kc, where c is the velocity of the jet. 



273-274J VibrcUions of a CylindriccU Jet 467 

where, as in Art. 210 (11), 

• ^^^ " 2^7 ! r "^ 2 (2« + 2) "^ 2 . 4 (2«' + 2) (2« + 4) "^ • • T '"^^"^^ 

Hence, writing 

<f> — BIf (kr) cos 80 cos fa , cos (o< + c), (14) . 

we have, by (4), 

l=.-B ^^' ^^^ cos «» cos fa . sin (o< + €) (15) 

To find the sum of the principal curvatures, we remark that, as an obvious 
consequence of Euler's and Meunier's theorems on curvature of surfaces, 
the curvature of any section differing infinitely little from a principal normal 
section is, to the first order of small quantities, the same as that of the 
principal section itself. It is sufficient therefore in the present problem 
to calculate the curvatures of a transverse section of the cylinder, and 
of a section through the axis. These are the principal sections in the 
undisturbed state, and the principal sections of the deformed surface will 
make infinitely small angles with them. For the transverse section the 
formula (6) applies, whilst for the axial section the curvature is — 3*C/3«* ; so 
that the required sum of the principal curvatures is 



l + l-'^-Kr + ^S). 






= 1 _ gMjL^ (jfc«c« + »> - 1) COS sd cm hi. Bin (at + e) (16) 

a aa* 

Also, at the surface, 

^ = ^ = — aBl, (ka) cos «d cos Xa; . sin ((rf + c) (17) 

p ot 

The surface-condition of Art. 265 then gives 

°*- w'*'°*-''-"-^' *'" 

For « > 0, cr' is positive ; but in the case {s = 0) of symmetry about the axis 
c^ will be negative if ha<\\ that is, the equilibrium is unstable for 
disturbances whose wave-length (^jk) exceeds the circumference of the jet. 
To ascertain the type of disturbance for which the instability is greatest, we 
require to know the value of ha which makes 

/o (ha) 

a maximum. For this Rayleigh finds i*a* = '4858, whence, for the wave- 
length of maximum instability, 

2ir/it = 4-508 x 2a. 

30—2 



468 Sfwrface. Waves [ohap. ix 

There is a tendency therefore to the production of bead-like swellings 
and contractions, of this wave-length, with continually increasing amplitude, 
until finally the jet breaks up into detached drops'^. 

275. This leads naturally to the discussion of the small oscillations of 
a drop of liquid about the spherical formf. We will slightly generalize the 
question by supposing that we have a sphere of liquid, of density p, 
surrounded by an infinite mass of other liquid of density p\ 

Taking the origin at the centre, let the shape of the common surface at 
any instant be given by 

r = a -i- i = a -i- iSn . sin (<rf + €), (1) 

where a is the mean radius, and S^ is a surface-harmonic of order n. The 
corresponding values of the velocity-potential will be, at internal points, 

^ = - ^ ^ -S, . COS (o< + «), (2) 

and, at external points, 

since these make ^ ^ "" ^ ^ "" ^' 

for r =^ a. The variable parts of the internal and external pressures at the 
surface are then given by 

p = .*.". + ^- — Sn . sin (of + c), y' = . . . — ^"TT ^" . sin (o< -i- €). ... (4) 

To find the sum of the curvatures we make use of the theorem of SoUd 
Geometry, that if A, ft, v be the direction-cosines of the normal at (x, y, z) to 
that surface of the family 

jP (aj, y, z) = const. 

which passes through the point, viz. 

XV 1 . 1 dX dfjL dv ,^. 

^^^"^ fii + ^ = ai + a^ + ai (s> 

* The argument here is that if we have a series of possible types of disturbance, with time* 
factors e'^ , e^ , e^ , . . ., where ai>a,>a3> . . ., and if these be excited simultaneously, the 
ampHtude of the first will increase relatiyely to those of the other components in the ratios 
^(«i-«t) ^ ^ *!-«») ^ ^ ^ ^ ^ The component with the greatest a will therefore ultimately predominate. 

The instability of a cylindrical jet surrounded by other fluid has been discussed by Rayleigh, 
"On the Instability of Cylindrical Fluid Surfaces," PhU. Mag. {$), t. zxxiy. p. 177 (1892) [Papers, 
t. iii. p. 694]. For a jet of air in water the wave-length of maximum instability is found to be 
6*48 X 2a. 

t Rayleigh, Ic. anU p. 466; Webb, Mess, of Math. t. ix. p. 177 (1880). 



274-275] Vibrations of a Globule 469 

Since the square of ([ is to be neglected, the equation (1) of the harmonic 
spheroid may also be written 

r-o + t., (6) 

where U = ^S^.an(at + e) (7) 

i.e. {n is A '0^ harmonic of degree n. We thus find 



r dy 



/*-!-^ + nS{..)- (8) 









whence 
1,1 2 n{n + l)y 2 . (n - 1) (n + 2) _ • ._ , , ,„, 

Substituting from (4) and (9) in the general surface-condition of Art. 265, 
we find 

.' = n(n+l)(«-l)(n + 2)^^ ^^^^J^^^^^^. (10) 

If we put p' = 0, this gives 

«r» = n(n-l)(n + 2)^,. (11) 

The most important mode of vibration is that for which n = 2 ; we then have 



cr«=: 



pa^' 



Hence for a drop of water, putting jTi = 74, p — 1, we find, for the frequency, 

a/2ir = 3'87a"* vibrations per second,. 

if a be the radius in centimetres. The radius of the sphere which would 
vibrate seconds is a » 2'47 cm. or a little less than an inch. 

The case of a spherical bubble of air, surrounded by liquid, is obtained 
by putting p » in (10), viz. we have 

a«=(n+l)(n-l)(n+2)^3 (12) 

r 

For the same density of the liquid, the frequency of any given mode is 
greater than in the case represented by (11), on account of the diminished 
inertia : cf. Art. 91 (7), (8). 



«- • 



CHAPTER X 



WAVES OF EXPANSION 



276. A TBEATISE on Hydrodynamics would hardly be complete without 
some reference to this subject, if merely for the reason that all actual fluids 
are more or less compressible, and that it is only when we recognize this 
compressibility that we escape such apparently paradoxical results as that of 
Art. 20, where a change of pressure was found to be propagated instofUaneously 
through a liquid mass. 

We shall accordingly investigate in this Chapter the general laws of 
propagation of small disturbances, passing over, however, for the most part, 
such details as belong more properly to the Theory of Sound. 

In most cases which we shall consider, the changes of pressure are small, 
and may be taken to be proportional to the changes in density, thus 

Ap = K . —, 
P 

where k (^ p dp/dp) is a certain coefficient, called the 'elasticity of volume.' 
For a given liquid the value of k varies with the temperature, and (very 
slightly) with the pressure. For water at 15® C, k = 2*045 x 10^® dynes per 
square centimetre. The case of gases will be considered presently. 

Plane Waves, 

277. We take first the case of plane waves in a imiform medium. 

The motion being in one dimension (x), the dynamical equation is, in the 
absence of extraneous forces, 

dt dx"" pdx^ pdpdx' ^ ' 

whilst the equation of continuity. Art. 7 (5), reduces to 

t + hl^)-^ (2) 

If we put p = po(l + «), (3) 



where c* 



276-278] Plane Waves 471 

where p^ is the density in the undisturbed state, a may be called the 'con- 
densation' in the plane x. Substituting in (1) and (2), we find, on the 
supposition that the motion is infinitely small, 

di'^J^Fx' ^ ^ 

!'--& <»' 

as above. Eliminating « we have 

=f=rgi (8) 

The equation (7) is of the form treated in Art. 170, and the complete 
solution is 

u =f{ct - a;) + J (c« + x), (9) 

representing two systems of waves travelling with the constant velocity c, 
one in the positive and the other in the negative direction of x. It appears 
from (5) that the corresponding value of 8 is given by 

C8^f{ct-x)-F{ct'{-x) (10) 

For a single wave we have u = ± cs, (11) 

since one or other of the functions/, F is zero. The upper or the lower sign 
is to be taken according as the wave is travelling in the positive or the 
negative direction. It is easily shewn in this case that the approximations 
involved in (4) and (5) are valid provided u is everywhere small compared 
with c. 

There is an exact correspondence between the above approximate theory and that of 
* long ' gravity- waves on water. If we write Tf/h for 8, and gh for ic/po, the equations (4) and 
(5), above, become identical with Art. 169 (3), (6). 

278. With the value of k given in Art. 276, we find for water at 15° C. 

c = 1430 metres per second. 

The number obtained directly by Colladon and Sturm* in their experiments 
on the lake of Geneva was 1437, at a temperature of 8*^ C. 

* Ann. de Chim. et de Phys. t, zxxvi ( 1S27). It may be mentioned that the velocity of sound 
in water contained in a tube ib liable to be appreciably diminished by the yielding of tiie wall. 
See Helmholtz, Fortschritte d. Physik, t. iv. p. 119 (184S) [Wiss. Abh. tip. 242]; Korteweg, 
Wied, Ann. t. v. p. 526 (1878); Lamb, Manch, Mem. t. xlii. No. 1 (1898). 



472 Waves of Expansion [chap, x 

In the case of a gas, if we assume that the temperature is constant, the 
value of ic is determined by Boyle's Law 

J=f, (1) 

viz. « = Po> (2) 



-y© '»> 



so that " ' _ 

This is known as the * Newtonian' velocity of sound*. If we denote by 
H the height of a 'homogeneous atmosphere' of the gas, we have po = gpoB^ 
and therefore 

o = (9H)i, (4) 

which may be compared with the formula (13) of Art. 170 for the velocity of 

*long' gravity- waves in liquids. For air at 0*^ C. we have as corresponding 

values 

Po = 76 X 13-60 X 981, po = '00129, 

in absolute o.G.S. units ; whence 

c = 280 metres per second. 
This is considerably below the value found by direct observation. 

The reconciliation of theory and fact is due to Laplace f. When a gas is 
suddenly compressed, its temperature rises, so that the pressure is increased 
more than in proportion to the diminution of volume ; and a similar state- 
ment applies of course to the case of a sudden expansion. The formula (1) is 
appropriate only to the case where the expansions and rarefactions are so 
gradual that there is ample time for equalization of temperature by thermal 
conduction and radiation. In most cases of interest, the alternations of 
density are exceedingly rapid ; the flow of heat from one element to another 
has hardly set in before its direction is reversed, so that practically each 
element behaves as if it neither gained nor lost heat. 

On this view we have, in place of (1), the 'adiabatic' law 

j'=C^y. (5) 

Po ^Po^ 
where, as explained in books on Thermodynamics, y is the ratio of the two 
specific heats of the gas. This makes 

K^yPo, (6) 

and therefore c = ^/f— ) = ViygB) , (7) 

♦ Principia, Lib. ii. Sect. viii. Prop. 48. 

t The usual referenoe is to a paper "Sur la vitesse du son dans Tair et dans l*eau," Ann, de 
Chim. et de Phy$, t. iii p. 238 (1816) [M4canique COesU, Livre 12»«, o. iii (1823)]. But Poisson 
in a memoir of date 1607 (see p. 479) refers to this explanation as having been already given by 
Laplace. 



278-279] Vdoeity of Sound 473 

If we put y » 1*402*, the former result is to be multiplied by 1*184, whence 

c s 3S2 metres per second, 
which agrees very closely with the best direct determinations. 

The oonfidenoe felt by phydoistB in ihe soundness of Laplace's view is so oomplete that 
it is now usual to apply the formula (7) in the inverse manner, and to infer the values of y 
for various gases and vapours from observation of wave-velooities in them. 

In strictness, a similar distinction should be made between the ^adiabatic' and 
* isothermal* coefficients of elasticity of a liquid or a solid, but practically the difierenoe 
is unimportant^ Thus in the case of water the ratio of the two volume-elasticities 
is calculated to be 10012t. 

The effects of thermal radiation and conduction on air-waves have been studied 
theoretically by Stokes ^ and Rayleighf. When the oscillations are too rapid for equaliza- 
tion of temperature, but not so rapid as to exclude communication of heat between adjacent 
elements, the waves diminish in amplitude as they advance, owing to the dissipation of 
energy which takes place in the thermal processes. The effect of conduction will be noticed, 
along with that of viscosity, in the next Chapter. 

According to the law of Charles and Dalton 

P^BpB, (8) 

where d is the absolute temperature, and £ is a constant depending on the 
nature of the gas. The velocity of sound will therefore vary as the square 
root of 0. For several of the more permanent gases, which have sensibly the 
same value of y, the formula (7) shews that the velocity varies inversely 
as the square root of the density, provided the relative densities be deter- 
mined under the same conditions of pressure and temperature. 

• 

279. The theory of plane waves can also be treated very simply by the 
Lagrangian method (Arts. 13, 14). 

If ^ denote the displacement at time t of the particles whose undisturbed 
abscissa is x, the stratum of matter originally included between the planes x 
and a; + Sx is at the time t-^-ht bounded by the planes 

a? + f and ^ +^ + (l + g|) S^> 
so that the equation of continuity is 

'>(i+i) = '>o' (1) 

* The value found by the most recent direct experiments. 

t Everett, XJniU and Physical Constants, 

X "An Examination of the possible effect of the Radiation of Heat on the Propagation of 
^onnd,'' Pha. Mag. (4), t. i. p. 306 (1861) [Papers, t. iiL p. 142]. 

I Theory of Sound, Art. 247. In a paper ** On the Cooling of Air by Radiation and Conduction, 
and on the Propagation of Sound/' PhU. Mag, (6), t. xlvii. p. 308 (1899) [Papers, t. iv. p. 376], 
Rayleigh concludes on experimented grounds that conduction is much more effective in this 
respect %h$n radiation. 



474 



Waves of Bxpansion 



[chap. X 



where po ^^ ^^^ densitj in the undisturbed state. Hence if s denote the 
'condensation' (/> — PoVpo* we have 

dj 

^"^ (2) 



« = — 



r3' 

ax 



The dynamical equation, obtained by considering the forces acting on 
unit area of the above stratum, is 



a«| dp 

'*«a<«~ dx' 



(3) 



These equations are exact, but in the case of small motions we may write 

p = Po + K8y (4) 

(5) 



and 

Substituting in (3) we find 



dx 






= c« 



dx^' 



(6) 



'i 



where c" = k/pq. The solution of (6) is the same as in Arts. 170, 277. 

280. The kinetic energy of a system of plane waves is given by 

T = \poiUuHxdydz (1) 

where u is the velocity at the point (a?, y, z) at time t. 

The calculation of the intrinsic energy requires a 
little care. The work done by unit mass in expanding 
through a small range, from the actual volume v to the 
standard volume v^, is given to the second order of 
small quantities by the expression 

as is obvious on inspection from Watt's diagram. 
Putting 

P = Vli + K8, Vq-V = 8Vq, (2) "" 

we have 

i (P + Po) (Vo - v) = Po (% - v) 4- i (2> - yo) {% - v) 

*= Po (vo -v) + ^ks^Vq (3) 

If we take the sum of the corresponding expressions for all the mass- 
elements of the system, the term y© (^o ~ ^) will disappear whenever the 
conditions are such that the total change of volume is zero. This being 
assumed, we have, for the work done by the gas contained in any given 
region, in passing from its actual state to the normal state, the expression 

W = ^Kfffs^dxdydz (4) 



V Vr 



279-281] Energy of Somid-Waves 475 

So far, no assumptioii has been made as to the precise manner in which the 
transition takes place ; this will affect the value of ic. It is only in the case 
of adiabatic expansion that the expression (4) can be identified with the 
intrinsic energy' in the strict sense of the term. When the expansion is 
isothermal^ the expression gives what is known in Thermodynamics as the 
* free energy.* 

In a progressive plane wave we have c8 = ±Uy and therefore T ^ W. The 
equality of the two kinds of energy, in this case, may also be inferred from 
the more general line of argument given in Art. 174. 

In the Theory of Sound special interest attaches, of course, to the case of 
simple-harmonic vibrations. If a be the amplitude of a progressive wave 
of period 27r/a, we may assume, in conformity with Art. 279 (6), 

i = a cos {kx — of + €), (6) 

where k = a/Cy and the wave-length is accordingly A «= 2ir/i. The formidae 
(1) and (4) then give, for the energy contained in a prismatic space of 
sectional area unity and length A (in the direction x), 

T-hW^ iPo<^«* A, (6) 

the same as the kinetic energy of the whole mass when animated with the 
maximum velocity oa. 

The rate of transmission of energy across unit area of a plane moving with the particles 
situate in it is 

© -^ = p(ra 8an(kx - ai + f) (7) 

The work done by the constant part of the pressure in a complete period is zero. For the 
variable part we have 

Ap = ks = - K^= Kka sin (kx - trt + t) (8) 

Substituting in (7), we find, for the mean rate of transmission of energy, 

\Kvka^ = \pQfT^a* X c (9) 

Hence the energy transmitted in any number of complete periods is exactly that cor- 
responding to the waves which pass the plane in the same time, as we should expect, since, 
c being independent of X, the group- velocity is identical with the wave- velocity (cf . Art. 237). 



Waves of Finite AmjUitude. 

281. If ;? be a function of p only, the equations (1) and (3) of Art. 279 
give, without approximation, 

dH_^d^ dH ... 

di* ~ p„* dp • dx* • ^^' 



476 Waves of Uxpansion [chap, x 

On the 'isothenaal' hypothesis that 

^ = f (2) 

this becomes 5^ »= — . ai,a (3) 

In the same way, the 'adiabatic' rdation 

w 



p - fp""" 



Po \P0' 

leads to ?L^ _ rPo gg* (K\ 



(•-e 



These exact equations (3) and (5) may be compared with the similar equation for *long' 
waves in a uniform canal, Art. 173 (3). 

It appears from (1) that the equation (6) of Art. 279 could be regarded as exact if the 
relation between p and p were such that 

p't'""*"' («) 

Hence plane waves of finite amplitude can be propagated without change of type if, and 
only if, 

i'-J'o = /»«c«(l-^) (7) 

A relation of this form does not hold for any known substance, whether at constant 
temperature or when free from gain or loss of heat by conduction and radiation*. Hence 
sound-waves of finite amplitude must inevitably undergo a change of type as they proceed. 

282. The laws of propagation of waves of finite amplitude have been 
investigated, independently and by difEerent methods, by Eamshaw and 
Riemann. It is proposed to give here a brief account of these investigations, 
referring for further details to the original papers, and to the very full 
discussion of the matter by Rayleighf. 

Riemann's method:): has already been applied in this treatise to the 
discussion of the corresponding question in the theory of 'long' gravity- waves 
on liquids (Art. 187). He starts from the Eulerian equations (1) and (2) of 
Art. 277, which may be written 

9« + „9«=_1^9p, (1) 

ot ox p dp ox 

di^'^Fx-'-^d-x <^^ 

* The relation would make p negative when p falls below a certain value. 
t Theory of Sound, 0. zi 

X **Ueber die Fortpflanzung ebener Loftwellen von endlicher Schwingungsweite/' Q6U, Ahk* 
t. viii. p. -43 riM&-9) {Wtfkt, 2** Aufl., Leipzig, 1892, p. 167]. 



281-282] Waves of Finite Amplitude 477 

Ifweput P=/(p) + t*, Q^f{p)-u, (3) 

where / (p) is a-s yet undetermined, we find, multipljdng (2) by /' (p), and 
adding to (1), 

dP dP_ ldpdp_ /// x3t* 

If we now determine / (p) so that 

</'(''»*=^.| <*> 

this may be written 

dP , dP -, , , dP ,^. 

8? + ^^=-^-^(^)^ "(^^ 

In the aame way we obtain 

The condition (4) is satisfied by 
Substituting in (5) and (6), we find 

Hence dP «= 0, or P is constant, for a geometrical point moving with the 
velocity ^ 

S-(|)'+«. « 

whilst Q is constant for a point whose velocity is 

i-(^)*+» <'») 

Hence, any given value of P moves forward, and any value of Q move& 
backward, with the velocity given by (9) or (10), as the case may be. 

These results enable us to imderstand, in a general way, the nature of 
the motion in any given case. Thus if the initial disturbance be confined 
to the space between the two planes a; = a, a; » 6, we may suppose that P 
and Q both vanish for a; > a and for x<h. The region within which P is 
variable will advance, and that within which Q is variable will recede, until 
after a time these regions separate and leave between them a space for 
which P = 0, Q » 0, and in which the fluid is therefore at rest. The original 



478 Waves of JEocpansion [chap, x 

disturbance lias thus been split up into two progressive waves travelling in 
opposite directions. In the advancing wave we have Q = 0, and therefore 

«=/0»), (11) 

so that both the density and the particle- velocity are propagated forwards at 
the rate given by (9). Whether we adopt the isothermal or the adiabatic 
law of expansion, this velocity of propagation will be found to be greater, the 
greater the value of p. The law of progress of the wave may be illustrated 
by drawing a curve with x as abscissa and p as ordinate, and making each 
point of this curve move forward with the appropriate velocity, as given by 
(9) and (11). Since those parts move faster which have the greater ordinates, 
the curve will eventually become at some point perpendicular to z. The 
quantities du/dx, dp/dx are then infinite ; and the preceding method fails to 
yield any information as to the subsequent course of the motion. Cf . Art. 187. 

283. Similar results can be deduced from Eamshaw's investigation*, 
which is, however, somewhat less general in that it applies only to a pro- 
gressive wave supposed already established. 

For simplicity we will suppose p and p to be oonnected by Boyle's Law 

P = c*P (1) 

If. we write y = a; -f £, so that y denotes the absolute oo-oidinate at time t of the particle 
whose undisturbed abscissa is x, the equation (3) of Art. 281 becomes 

S-3/©' m 

This is satisfied by & " ■^ (X) ('' 

-^ M2)}"-/(2)' '« 

Hence a first integral of (2) is ^ = C' ± c log ^ (6) 

ot ox 

To obtain the * general integral' of (5) we must eliminate a between the equations 

y = or + (C± c log a) « + (a)\ 

= or ± c/ + a<^' (a), J ^"^' 

where ^ is arbitrary. Now ^ = ^, 

so that, if u be the velocity of the particle x, we have 

« = ^ = C±clog^« (7) 

01 p 

On the outskirts of the wave we shall have u = 0, p = pq. It follows that C = 0, and 
therefore 

P=Po«'^'''' (8) 

* "On the Mathematical Theoiy of Sound," PkU. Trans, t. cl. p. 133 (1868). 
t See Forsyth, Differential Equations, c. iz. 



282-284] Waves of Finite Amplitude 479 

Hence in a pn^ressive wave p and u must be connected by this relation. If this be 
satisfied initially^ the function ^ which occurs in (6) is to be determined from the conditions 
at time ^ = by the equation 

<t>'{polp)= -x (9) 

To obtain results independent of the particular form of the wave, consider two particles 
(which we will distinguish by sufi&xee) so related that the value of p which obtains for the 
first particle at time t^ is found at the second particle at time t^. The value of a (= pjp) 
is the same for both, and therefore by (6), with C = 0, 

Vi- yi = «(«8 - a:i) ± c (t^ - t^) log a,\ 

The latter equation may be written -— = T c — , (11) 

^ Pq 

shewing that the value of p is propagated from particle to particle at the rate p/pf, , c. The 
rate of propagation in apace is given by 

^= Tc±c log a= qPc + M (12) 

This is in agreement with Riemann*s results, since on the present hypothesis of isothermal 
expansion (dpjdpY = c. 

For a wave travelling in the positive direction we must take the lower signs. If it be 
one of condensation (p > po), u is positive, by (8). It follows that the denser pckrts of the 
wave are continually gaining on the rarer, and at length overtake them ; the subsequent 
motion is then beyond the scope of our sknalysis. 

Eliminating x between the equations (6). and writing for c log a its value -t^ we find, 
lor a wave travelling in the positive direction, 

y=(c + u)t + F (a), (13) 

where J* is an arbitrary function. In virtue of (8) this is equivalent to 

n = f{y-(c + u)t} (U) 

This formula is due to Poisson*. Its interpretation, leading to the same results as above, 
for the mode of alteration of the wave as it proceeds, forms the subject of a paper by 
Stokes t. 

284. The conditions for a wave of permanent type have been investigated 
in a very simple manner by Bankine|. 

Let Ay B he two points of an ideal tube of unit section drawn in the 
direction of propagation, whicb is (say) that of x positive, and let the values 
of the pressure, the density, and the particle-velocity at A and B be 
denoted by p-^, pi, u^ and p^, p^, u^, respectively. 

If, as in Art. 175, we impress on everything a velocity c equal and opposite 
to that of the wave, we reduce the problem to one of steady motion. Since 

* "M^moire sur la th^orie du son,'* Joum. de VicoU Polytechn, t. viL p. 367 (1807). 

t "On a Difficulty in the Theory of Sound," PhU, Mag. (3), t. xxiii. p. 349 (1848) [Paptrs, 
t. ii. p. 51]. 

} "On the Thermodynamic Theory of Waves of Finite Longitudinal Disturbance," Phil, 
Trans, t. clx. p. 277 (1870) [Papers, p. 630]. 



480 Waves of Eoqparmon [ohap. x 

the same amount of matter now crosses in unit time each section of the 
tube, we have 

Pl (C - Ui) = Pj, (C - Wj) = w (1) 

say ; where m denotes the mass swept past in unit time by a plane moving 
with the wave, in the original form of the problem. This quantity m is called 
by Bankine the 'mass- velocity' of the wave. 

Again, the total force acting on the mass included between A and B is 
p^ - p^, in the direction BA, and the rate at which this mass is gaining 
momentum in the same direction is 

in(c — Uj) — m(c — u^). 

Hence P2 "" ?i = ^ (^t "" ^1) (2) 

Combined with (1) this gives 

l>i + — = P« +— (3) 

Pi P2 

Hence a wave of finite amplitude could not be propagated unchanged except 
in a medium such that 

p-\ = const (4) 

This conclusion has already been arrived at, in a different manner, in Art. 281. 
It may be noticed that, if we write v = 1/p, the relation (4) is represented on 
Watt's, diagram by a straight line. 

If the variation of density be slight, the relation (4) may, however, be 
regarded as holding approximately for actual fluids, provided tn have the 
proper value. Putting 

p = Po (! + «)» P = Po'^KS, m=^p^c (5) 

we find c* = — , (6) - 

/>o 

as in Art. 277. 

The fact that in actual fluids a progressive wave of finite amplitude 
continually alters its type, so that the variations of density towards the front 
become more and more abrupt, has led various writers to speculate on the 
possibility of a wave of discontinuity, analogous to a 'bore' in water-waves 
(cf. Art. 187). 

It was shewn, first by Stokes*, and afterwards by several other writers, 
that the conditions of constancy of mass and of constancy of momentum can 
both be satisfied for such a wave. The simplest case is when there is no 
variation in the values of p and u except at the plane of discontinuity. If, 

* 2.e. ante p. 479. 



m 



284] Condition for Permanent Type 481 

in the preceding argument, the sections A, B be taken, one behind, and the 
other in front of this plane, we have, by (3), 

= (!^*-'"^«)* <') 

c-„. = ^ = (2l^P«.ei)*. (8) 

Pa \P1-P2 P%^ 

and ^, > ^, = ^ _ ^ = i ((PLll.?^)iPx JI_£l))* (9) 

P% Pi \ Pi Pi / 

The upper or the lower sign is to be taken according as pi is greater or less 
than P2, i.e. according as the wave is one of condensation or of rarefaction. 
The results involve differences of velocity, as we should expect, since any 
uniform velocity of the whole medium may be superposed. 

We may assume, for instance, that the quantities f^, p%,u^, which define 
the condition of the medium ahead of the wave, are given arbitrarily; also 
that the density p^ of the air in the advancing wave is prescribed. Further, 
some definite relation between p^, p^ and p^, p2, based on physical considera- 
tions, is presupposed. The remaining quantities m, c, u^ are then determined 
by (7), (8), (9). 

These results are, however, open to the criticism* that in actual fluids 
the equation of energy cannot be satisfied consistently with (1) and (2). 
Calculating the excess of the work done per unit time on the fluid entering 
the space AB at B over that done by the fluid leaving at A, and subtracting 
the gain of kinetic energy, we obtain 

Pt (c - uz) -Pi(c- Ui) - \m {(c - u^Y - (c - u^Y), 
or y^Uj - p^U2 - im (wi* - u^^), 
or i {pi + p^) K - w,), (10) 

these forms being equivalent in virtue of the dynamical equation (2). The 
corresponding result per unit mass is obtained by dividing by m. If we 
substitute for u^ — Wg from (1) or (9), we obtain 

i {Pi + Pi) {'^i - t^i), (11) 

where v is written for 1/p. 

If the two states of the medium be represented by two points -4, B on 
Watt's diagram, the expression (11) is equal to the area included between 
the straight Une AB^ the axis of v, and the ordinates ol A^ B. If the 
transition from £ to ^ be effected without gain or loss of heat, the points 
in question will lie on the same 'adiabatic curve,' and the gain of intrinsic 
energy will be represented by the area included between this curve, the 
axis of Vy and the extreme ordinates. For an actual gas, the adiabatic is 

* Rayleigh, Theory of Sound, Art. 253. The comparison with Art. 187 ante is interesting. 
L. H. 31 



482 Waves of Expansion [chap, x 

concave upwards ; and the latter area is accordingly less (in absolute value) 
than the former. If we have regard to the signs to be attributed to the 
areas, we find that for a wave of condensation (v^ < v^ the work done on the 
medium is more than is accounted for by the increase of the kinetic and 
intrinsic energies ; whilst in a wave of rarefaction {v^ > v^ the work given 
out is more than the equivalent of the apparent loss of energy. 

It appears that the equation of energy cannot be satisfied for discon- 
tinuous waves, except in the case of a hypothetical medium whose adiabatic 
Unes are straight. This is identical with the condition already obtained for 
permanency of type in continuous waves. 

In the above investigation no account has been taken of dissipative 
forces, such as viscosity and thermal conduction and radiation. Practically, 
a wave of discontinuity would imply a finite difEerence of temperature 
between the portions of the fluid on the two sides of the plane of discontinuity, 
so that, to say nothing of viscosity, there would necessarily be a dissipation 
of energy due to thermal action at the junction. Whether, when this is 
allowed for, the relation between the two states can be reconciled with the 
equation of energy is a physical question into which we do not enter*. It 
would appear that the possibility of a discontinuous wave of rarefaction 
is in any case excluded, since (as may easily be shewn graphically) this 
would involve a loss of 'entropy' in an irreversible process. 

Some reference should however be made to investigations in which the 
transition from one uniform state to another is supposed to be continuous, 
though possibly very rapid. The admissible form of a wave of permanent, 
type, when thermal conduction is taken into account, was discussed by 
Rankine (i.e.). Rayleigh, in a recent important examination of the whole 
subjectf , has considered also the influence of viscosity. It appears that the 
wave, as already stated, must necessarily be one of condensation, and that 
if the ratio of the uniform pressures in front and rear of the wave differs 

* In some inyestigatioiiB by Hugoniot, which are expounded by Hadamard in his Lecons sur 
la propcigation des ondes et les iqnations de Vhydrodynamique, Paris, 1903, the argument given in 
the text is inverted. The possibility of a wave of discontinuity being assumed, it is pointed out 
that the equation of energy will be satisfied if we equate the expression (10) to the increment of 
the intrinsic energy (for which see Art. 11 (8)). On this ground the formula 

i iPi +Pt) (vt - Vi) = —i (Pi^i -Pt^t) 

is propounded, as governing the transition from one state to the other: "Telle est la relation 
qu'Hugoniot a substitu6e k [pv^ = const.] pour exprimer que la condensation ou dilatation brusque 
se fait sans absorption ni d6gagement de chaleur. On lui donne aotuellement le nom de hi adia- 
hatique dynamiqtie, la relation [pv^ = const.], qui convient aux changements lents, ^tant d^sign^ 
sous le nom de lot adiabatique statique *' (Hadamard, p. 192). But no physical evidence is adduced 
in support of the proposed law. 

t "Aerial Plane Waves of Finite Amplitude," Proc, Roy. Soc. A, t. Ixxxiv. p. 247 (1910) 
[Papers, t. v. p. 673]. 



284-285] Spherical Waves 483 

appreciably from unity the transition is practically efEected in a space which 
is very minute, so that the circumstances closely approach those of a discon- 
tinuity*. 

Spherical Waves. 

285. Let us next suppose that the disturbance is symmetrical with 
respect to a fixed point, which we take as origin. The motion is necessarily 
irrotational, so that a velocity-potential <f> exists, which is here a function 
of r, the distance from the origin, and t, only. If as before we neglect the 
squares of small quantities, we have by Art. 20 (3) 

dp _ 3^ 
J ^ dt • 

In the notation of Arts. 276, 277 we may write 



/' 



/?-/t-«-- 



P J Po 
whence ^** ^ ^ (^) 

To form the equation of continuity we remark that, owing to the difference 
of flux across the inner and outer surfaces, the space included between the 
spheres r and r + 8r is gaining mass at the rate 

i(4«r«p|)8r. 

Since the same rate is also expressed by dp/dt . 47rr*Sr we have 



"l-alK) <^) 



This might also have been arrived at by direct transformation of the general 
equation of continuity, Art. 7 (5). In the case of infinitely small motions, it 
becomes 

dt^7^d^Vd^j' (^) 

whence, substituting from (1), 

dt^ r^drV drj ^*^ 

This may be put into, the more convenient form 

a^2 -^ gV» > W 

so that the solution is 

r(f> ^f(r -ct)']-F(r + ct) (6) 

* Similar conclusions were arrived at independently by G. I. Taylor, "The Conditions 
Necessary for Discontinuous Motion in Gases," Proc. Boy, Soc, A, t. Ixzxiv. p. 371 (1910). 

31—2 



484 Waves of Exparmon [chap, x 

Hence the motion is made up of two systems of spherical waves, travelling, 
one outwards, the other inwards, with velocity c. Considering for a moment 
the first system alone, we have 

c«= -^f {r-ct), 

which shews that a condensation is propagated outwards with velocity c, but 
diminishes as it proceeds, its amount varying inversely as the distance from 
the origin. The velocity due to the same train of waves is 

As r increases the second term becomes less and less important compared 
with the first, so that ultimately the velocity is propagated according to the 
same law as the condensation. 

We notice that whenever diverging or converging waves are alone present 
we have 

J|:W=^^; (7) 

this corresponds to Art. 277 (11). 

For some purposes the formula for a system of divergent waves is more 
conveniently written 

4^<f>^f(t-^^ (8) 

Since this makes 

liiiWor-47rr«^l =/(<), (9) 

the system in question may be regarded as due to a source of strength /(<) 
at the origin; cf. Art. 196. 

It follows from (1) that 

8dt = 0y (10) 



/' 



provided the initial and final values of (f> both vanish. This will be the case 
whenever the source / (t) is in action only for a finite time. The fact that 
a diverging spherical wave must necessarily contain both condensed and 
rarefied portions was first remarked by Stokes*. Cf. Art. 197. 

As in the case of plane progressive waves (Art. 280), the energy of 
a system of divergent spherical waves is half kin«tic and half potential. 
This follows from the general argument of Art. 174, and may be verified 
independently as follows. We have, identically, 



-m-m-^^- 



♦ "On Some Points in the Received Theory of Sound," Phil, Mag. (3), t. xxxiv. p. 62 (1849) 
[Papers, t. ii. p. 82]. See also Rayleigh, Theory of Sound, Art. 279. 



286-286] Spherical Waves 486 

If we write ?= — ^> '^*'~'^' ^^^^ 

this gives, by (7), in the case of a divergent wave-Bystem, 

"CO /* 00 

Hence / yq^.im^dr = yc^s^.i^nr^dr, (12) 

Jo Jo 

if r<f>^ vanishes at the inner and outer boundaries of the system*. 

286. The determination of the functions / and F in (6), in terms of the 
initial conditions, for an unlimited space, can be e£Eected as follows. 

Let us suppose that the distributions of velocity and condensation at 
time t = are determined by the formulae 

<f> = ^(r), ^ = X('-) (13) 

where tji, x are arbitrary functions. Comparing with (6), we have 

/(2) + J'(z) = 2^(2), (U) 

-/'(2) + J'(z) = ?X(2). 

the latter of which gives on integration 

-f{z) + F{z) = lf' zx(z) + C (15) 

Again, the condition that there is no creation or annihilation of fluid at the 
origin gives 

f(-z) + F(z) = (16) 

The formulae (14) and (15) determine the functions/ and F for positive values 
of z; and (16) then determines /for negative values of zf. 

The final result may be written 

^<l>= ^ (r - ct)^ {r - ct) -]- ^ (r -{- ct) ^ (r + ct) -{- ^ j zx (z)dz, 



(17) 



or 



1 f ^+** 
r<f> ^ - l(ct - r)tp {ct - r) + ^ (ct + r)>l, (ct + r) + ^ zx{z)dz, 

(18) 

according as r is greater or less than ct. 



* Proc. Lond. Math, Soc. t. xxxv. p. 160 (1902). 
t Rayleigh, Theory of Sound, Art. 279. 



486 Waves of Uocpansion [chap, x 

Aa a very simple example we may suppose that the air is initially at rest, and that the 
initial disturbance consists of a um'form condensation Sq extending through a sphere of 
radius a. We have then yjt (z) = 0, whilst x (2) = c**q or according asz ^ a. At a distance 
r(> a) from the origin, the motion will not begin until i = (r - ayc^ and will cease when 
t = (r -\- a)/c. For intermediate instants we shall have 

r0 = iMo {a« - (r - c/)}2, (19> 

8 T " Ct 

and thence — = — s — (20) 

The disturbance is now confined to a spherical shell of thickness 2a; and the condensation 
s is positive through the outer half, and negative through the inner half, of the thickness. 

We shall require, shortly, an expression for the value of </) at the origin, 
for all values of t, in terms of the initial circumstances. We have, by (6) 
and (16), 

[9Jr«o = hmr=o 

T 

or, by (14) and the*consecutive equation, 

[^]r=0 = |.«^ (Ct) + tx (Ct) (21) 



General Equation of Sound-Waves. 

287. We proceed to the general case of propagation of expansion-waves. 
We neglect, as before, small quantities of the second order, so that the 
dynamical equation is, as in Art. 285, 

«"-f (» 

Also, writing p = p^ (1 -f- «) in the general equation of continuity, Art. 7 (5), 
we have, with the same approximation, 

dt ~ dx* ^ dy* '^ dz^ ^ ' 

The elimination of s between (1) and (2) gives 

dfi ~ \dx* ^ dy* ^ dzy' ^ ' 

or, in OUT former notation, 

^ = c»VV (4) 

Since this equation is linear, it will be satisfied by the arithmetic mean of 
any number of separate solutions 4>i, <f>%, <f>^, • , . * As in Art. 38, let us 
imagine an infinite number of systems of rectangular axes to be arranged 



286-287] General EqucUion of Sound- Waves 487 

uniformly about any point P as origin, and let <^i, ^j* ^s* • • • b® ^^® velocity- 
potentials of motions which are the same with respect to these systems as the 
original motion <}> is with respect to the system x, y, z. In this casejthe 
arithmetic mean (^, say) of the functions <l>i,(f>2,<f>3, ... will be the velocity- 
potential of a motion symmetrical with respect to the point P, and will 
therefore come imder the investigation of Art. 286, provided r denote the 
distance of any point from P. In other words, if ^ be a function of r and t, 
defined by the equation 

*=^/jV<i«', (^^ 

where </> is any solution of (4), and 8m is the solid angle subtended at P by 
an element of the surface of a sphere of radius r having this point as centre, 
then 

dt^ ^"^ dr^ ^^' 

Hence r4> =f(r - d) -\- F {r ■\- ct) (7) 

The mean value of (f> over a sphere having any point P of the medium 
as centre is therefore subject to the same laws as the velocity-potential of 
a symmetrical spherical disturbance. We see at once that the value of <f> 
at P at the time t depends on the means of the values which ^ and d^jdt 
originally had at points of a sphere of radius ct described about P as centre, 
so that the disturbance is propagated in all directions with imiform velocity c. 
Thus if the original disturbance extend only through a finite portion E of 
space, the disturbance at any point P external to Z will begin after a time 
T'jjc, will last for a time {^g — ri)/c, and will then cease altogether; r^, r^ 
denoting the radii of two spheres described with P as centre, the one just 
excluding, the other just including S. 

To express the solution of (4), already virtually obtained, in an analytical 
form, let the values of (f> and 3^/3^, when i = 0, be 

<f> = tP(x,y,z), ^ = xi^^y>^) (8) 

The mean values of these functions over a sphere of radius r described about 
{x, y, z) as centre are 

^ = v- Ij ^ (a? 4- ir, y + wr, z + nr) dm, 
^^^ij^^^'^ ^^' y + wir, z + nr) dm, 

* This result was obtained, in a different manner, by Poisson, '*M6moire sur la th^rie du 
son," Joum, de V^cdU Polytechn. t. vii. pp. 334-338 (1807). The remark that it leads at onoe 
to the complete solution of (4) is due to Liouville, Journ. de Math. t. i. p. 1 (1856). 



+ 



- -Mi t-IIM '!)«»*• 



488 Waves of Expa^ision [chap, x 

where Z, w, n denote the direction-cosines of any radius of this sphere, and 
8w the corresponding elementary solid angle. If we put 

I = Bind cos a>, m = sin sin a>, n = cos 0, 

we shall have Sro = sin 0i08a). 

Hence, comparing with Art. 286 (21), we see that the value of (f> at the point 
(x, y, z), at any subsequent time t, is 

(f> = —^ ,t jlil/{x -{- ctsinO cos a>, y -{- cteiii0 sin a>, z + ci cos 0) sin 0d0dot} 

J- JlX (x-{- ct sin cos a>, y + ci sin fl sin co, 2 + c^ cos 5) sin 0d0da)y 

(9) 

which is the form given by Poisson* 

288. The expression for the kinetic energy of the fluid contained in any 
given region is 

-=*^-///{(S)'-(|)'-(l)'}"»* '-> 

T. dT 

Hence -^ 

where ^ stands for d(f>/dL By Green's Theorem (Art. 43), this may be put 
in the form 

^l-- - Pojj'i'^dS- p,jjJ4>V*<kdxdydz 
= - po//<^g dS - P^,fjji>i>dxdydz. 
Hence if W = ^k jjjs* dxdydz = i ^ fjU* dxdydz (2) 

wehave | (7+ TF) = - po//«^|^rfS (3) 

We have seen (Art. 280) that, subject to a certain condition, W represents 
the intrinsic energy. 

The complete interpretation of (3) may be left to the reader. In various 
important cases, e.g. when